Abstract |
A NEW COMBINATORIAL INTERPRETATION OF THE MULTIPLE BINOMIAL COEFFICIENTS A Thesis by RACHEL LANDERS Submitted to the Office of Graduate Studies of Texas A&M University-Commerce in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE May 2015 A NEW COMBINATORIAL INTERPRETATION OF THE MULTIPLE BINOMIAL COEFFICIENTS A Thesis by RACHEL LANDERS Approved by: Advisor: Hasan Coskun Committee: Padmapani Seneviratne Stuart Anderson Rebecca Dibbs Yelin Ou Head of Department: Tingxiu Wang Dean of the College: Brent Donham Dean of Graduate Studies: Arlene Horne !iii Copyright © 2015 Rachel Landers !iv ABSTRACT A NEW COMBINATORIAL INTERPRETATION OF THE MULTIPLE BINOMIAL COEFFICIENTS Rachel Landers, MS Texas A&M University-Commerce, 2015 Advisor: Hasan Coskun, PhD Multiple qt-binomial coefficients, the n-dimensional analog of the classical binomial coefficients, is an active research topic in the field of combinatorics. This project investigates the combinatorial interpretation and properties of the qt-binomial coefficients in the two dimensional case. The result will be useful in not only the field of combinatorics but also number theory, statistics, computer science and other fields where the sequences of binomial coefficients or q-binomial coefficients are used. !v ACKNOWLEDGEMENTS I’d like to thank Josh for the late-night ice creams runs; Trini for being there when my nerves are too much to handle; the Mathematics Department of Texas A&M University- Commerce for always being so supportive of their students; and finally, Dr. Coskun for his contributions and advice. !vi TABLE OF CONTENTS LIST OF FIGURES ..................................................................................................................... vii CHAPTER 1. INTRODUCTION ...................................................................................................... 1 2. BACKGROUND ........................................................................................................ 4 Classic Binomial Coefficients ............................................................................. 4 q-Binomial Coefficients ...................................................................................... 8 qt-Binomial Coefficients ...............................................................................… 15 Common Properties ........................................................................................... 18 3. THE PRINCIPAL THEOREM ................................................................................. 25 4. A NEW COMBINATORIAL INTERPRETATION IN TWO DIMENSIONS ......... 29 External Factors ................................................................................................. 31 Internal Factors .................................................................................................. 36 A Complete Example .....................................................................................… 42 5. FUTURE RESEARCH ............................................................................................. 51 REFERENCES .......................................................................................................................... 52 VITA ........................................................................................................................................ 56 !vii LIST OF FIGURES FIGURE 1. The Pascal Triangle ........................................................................................................... 6 2. The lattice paths of ! ................................................................................................. 12 3. The Ferrer diagram of λ=(5, 2, 1) .................................................................................... 12 4. All lattice paths for λ=(10, 8) and μ=(5,2) ...................................................................... 30 5. The external non-q-Pochammer factors and their weight distributions for λ=(10, 8) and μ=(5,2) ......................................................................................................................31 6. The external q-Pochammer factors and their weight distributions for λ=(10, 8) and μ=(5,2) ............................................................................................................................. 35 7. t−ρ(μ)q−ρ(μ)σ(λ) for λ=(10, 8) and μ=(5,2) ...................................................................... 38 8. (t)ρ(μ) for λ=(10, 8) and μ=(5, 2) ..................................................................................... 38 9. (qδ(μ)+1)ρ(μ) for λ=(10, 8) and μ=(5, 2) .......................................................................... 39 10. (qμ2−λ2)ρ(μ) for λ=(10, 8) and μ=(5, 2) ............................................................................ 39 11. (qρ(μ)+1)δ(μ) for λ=(10, 8) and μ=(5, 2) .......................................................................... 41 12. (t)δ(μ) for λ=(10, 8) and μ=(5, 2) ..................................................................................... 41 5 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ q !viii 13. (q−δ(λ))δ(μ) for λ=(10, 8) and μ=(5, 2) ........................................................................... 42 14. The external non-q-Pochhammer factors for λ=(4, 2) and μ=(3, 1) ................................ 43 15. The external q-Pochhammer factors for λ=(4, 2) and μ=(3, 1) ....................................... 44 16. The first lattice path for λ=(4, 2) and μ=(3, 1) ................................................................ 47 17. The second lattice path for λ=(4, 2) and μ=(3, 1) ............................................................ 49 18. The third lattice path for λ=(4, 2) and μ=(3, 1) ............................................................... 50 1 Chapter 1 INTRODUCTION The binomial coecients are a special sequence of numbers with many applied uses in various fields of math [40]. The generalizations of the binomial coecients in the q- and qt-analogs has become a hot topic in the field of combinatorics. Progress has been made within recent years in the study of the qt-binomial coecients (which will be further discussed in Section 2), but there still remains much to be discovered. Of these gaps in knowledge, the near absence of combinatorial interpretations of the qt-binomial coecients is perhaps the most conspicuous. Combinatorial interpretations of the classic binomial coecients are plentiful, and the q-binomial coecients already have multiple interpretations (most of which parallel those of the classic case). However, prior to this study only one combinatorial interpretation of the qt-binomial coecients existed–that of A. Okounkov, ✓ ! μ ◆ = P⇤ μ(q!; q, t) P⇤ μ(qμ; q, t) (1) which, unlike the combinatorial interpretation produced in this study, requires investigation of the properties of every square in every reverse Young tableau of μ. Because even for small partitions the number of possible reverse Young tableau can quickly reach tens of thousands, Okounkov’s interpretation can be very lengthy and extensive. Even by computer, the time for calculation using Okounkov’s method can range from seconds to minutes. The combinatorial interpretation introduced in this paper–presented as a Corollary in Section 4–drastically cuts the amount of calculation and time needed to find the qt-binomial coecients. H. Coskun defines the most general form of the qt-binomial coecients so that where z 2 C is an n-tuple of a complex number or a partition of an integer, q, t 2 C, and μ a partition of at most n parts, ✓ z μ ◆ q,t = q μ t2n(μ)+(1−n) μ (qtn−1)μ Y 1i wj . The number of entries to the left of i satisfying j > i. Let Gn be the set of all permutations of a set S. Further, let inv(w) denote the sum of all inversions of wn [37]. Then ✓ n k ◆ q = X w2Gn qinv(w) (23) Proof. A combinatorial proof is provided by Stanley on page 37 [37]. A demonstration of the inversion interpretation is provided below. Example 3. {3,2,1} has 0 descents. {3,1,2} has 1 descent. {2,3,1} has 1 descent. {2,1,3} has 2 descents. {1,3,2} has 2 descents. {1,2,3} has 3 descents. This corresponds to *4 2 + = 1+2q + 2q2 + q3 The tiling interpretation is of interest as well [2]. Let a board of length n using k green squares and n − k red squares assign to each square a weight qw. Let Tn,k be the set of all tilings of a board described as so using exactly k green squares and n − k red squares. Let qwT be the weight of tiling T 2 Tn,k where wT is calculated by assigning to each red square a weight of 1 and each green square the weight s so that s is equal to the number of red squares to the left of that green square in the tiling and then multiplying the weight qs of all the green squares. The q-binomial coecient *n k + q is created by summing the weights of all these tilings, i.e., 15 Lemma 10 (Tiling Interpretation). ✓ n k ◆ q = X Tn,k qwT (24) Proof. There exists a bijection between the tiling interpretation and the partition interpre-tation [2]. qt-Binomial Coecients Finally, the discussion arrives at the qt-binomial coecients which is the serious around which the results of this study revolve. Not much is known about these series of numbers and there exists little literature compared to that of the classic and q-binomial coecients. Information which is known and is relevant to this study are discussed here. The qt-binomial coecients add yet another parameter to the binomial coecients–the t parameter. By allowing the t parameter to approach 1, the qt-binomial coecients reduce to the q-binomial coecients which in turn reduce to the classic binomial coecients when q approaches 1. The currently known formulas for the qt-binomial coecients, which were derived using the Macdonald [29] polynomials, both rely heavily on the Young diagrams and Young tableau which are defined below. A Young diagram is a finite collection of left-justified boxes so that the nth entry in a partition = (1,2, . . . ,k) is the length of the nth row in the diagram. A reverse Young tableau is a Young diagram with positive integers from the set 1, 2, . . . ,k in each block such that the integers weakly decrease in the rows but strictly decrease in the columns [41]. When the squares are replaced with dots this is called a Ferrer’s diagram [39]. See Figure (3) for a demonstration of the Ferrer’s diagram. The definition constructed by H. Coskun [15,16] is the most general form of the binomial coecients in the sense that it is takes complex ntuples for its arguments. The following 16 partition statistics are used in this definition [15, 16]: (a)n = (a; q)n = Qn−1 i=0 (1−aqi), (a)! = (a; q, t)! = Qn i=1(at1−i; q)!i , μ = Pn i=1 μi, n(μ) = Pn i=1(i − 1)μi, and n(μ0) = Pn i=1 #μi 2 $ . Thus, the qt-binomial coecients as given by H. Coskun [15,16] is defined as the following. Definition 7. Where z 2 C is an n-tuple of a complex number or a partition of an integer,q, t 2 C and μ a partition of at most n parts such that ✓ z μ ◆ q,t = q μ t2n(μ)+(1−n) μ (qtn−1)μ Y 1i< >: 1−qa(s)tl(s)+1 1−qa(s)+1tl(s) 1 (31) and a(s) = !i j, a0(s) = j 1, l(s) = !j i, and l0(s) = i 1. Note that the summation runs over all possible reverse Young tableau; however, the set of all possible reverse Young tableau can be very large. In addition, the Okounkov interpretation must use statistics for each square in the reverse Young tableau. For this reason, it quickly becomes impossible to calculate the qtbinomial coecients using Okounkov’s formula without the use of a computer and even then can be lengthy to compute for larger partitions. As will be seen, the combinatorial interpretation presented in this paper relies on less computation based on characteristics of the reverse Young tableau which results in a more ecient method of computing the qt-binomial coecients. 18 0.0.2 Example Using The Interpretation as Provided By A. Okounkov An example using Definition (25) and partitions = (4, 2) and μ = (3, 1) is discussed here so that eciency can be compared between Definitions (25), (8), and (1). Despite the brevity of the Young tableau of = (3, 1), the number of reverse Young tableau which fit into this Young tableau is 15. Using code the author defines in Wolfram Mathematica, the following result is found: ✓ μ ◆ = (1 + q)(1 + q4t)(1 q q2t + qt2 + q2t2 + q3t2) t(1 + qt)(1 + q3t) (32) This result is the same as found in Formula (27);yet the time it took for Wolfram Math-ematica to compute the answer averages approximately 40.0 seconds in comparison to the approximately 0.02 seconds it takes to compute the same and μ using Definition (25). Common Properties Because the q- and qt-binomial coecients are a generalization of the classic binomial coecients, all three sequences share most of the known properties. The sum of each row in Pascal’s Triangle is a power of 2. In other words, Lemma 11. Xn k=0 ✓ n k ◆ = 2n (33) Proof. Algebraically, the proof of this quite easily follows from the Binomial Theorem. If a = 1 and b = 1, the property is achieved [17]. Combinatorially, the proof is done by showing that both sides of the property count the number of subsets of an n-element set S. First observe that for some k = 0, 1, 2, . . . ,n, every subset of S is a k-element subset of S. Because $n k % is the number of k-element subsets of S, it follows that $n 0 % + $n 1 % +· · ·+ $n n % is the number of subsets of S. The number of subsets of S can also be counted by observing that since S has n elements, when forming subsets of S each of the n elements have either one of two 19 fates: membership in the subset or exclusion from the subset. Thus, by the multiplication principle, there are 2n ways subsets of S can be formed [9]. The q-analogue of Lemma (11) is not found in the literature. However, by using its qt-analogue, it is easily reduced to its q-analogue form. Lemma 12. Y ( 1 + qk) = qμ ✓ μ ◆ q (34) Proof. Because the qt-binomial coecients reduce to the q-analogue of the binomial coe- cients as t approaches one, the same holds true for the properties. Therefore, to obtain the q-analogue of the property simply allow t to approach 1. The qt-analogue of the previous classic binomial coecient property is: Lemma 13. (1)= X μ✓qn(μ0)t−n(μ) ✓ μ ◆ qt (35) Proof. There exists an algebraic proof of this property in the qt-analogue, but since it is not combinatoric it is not included [16]. The property that previous terms in the sequence surrender subsequent terms [21] is known as the recursive property. Lemma 14 (Recursive Property of the Classic Binomial Coecients). ✓ n k ◆ = ✓ n 1 k 1 ◆ + ✓ n 1 k ◆ (36) The recursive property [36] of the classic binomial coecients is easily demonstrated in Pascal’s Triangle (Figure 1). Notice that the sum of any two consecutive integers in the same row surrender the integer below and between them. The combinatorial proof [25] is simple. Proof. Let A be a set of n objects. From A select one element, say x. All k-element subsets of A are then divided into two disjoint groups according to whether or not these contain x. 20 Because all sets containing x can choose k − 1 elements from the set A/{x}, which consists of n − 1 elements, there are !n−1 k−1 " subsets containing x. Because the subsets not containing x are able to pull elements from the n − 1 elements remaining in A which are not x, there are !n−1 k " subsets not containing x. Lemma 15 (Recursive Property of the q-Binomial Coecients). ✓ n + 1 k ◆ q = qk ✓ n k ◆ q + ✓ n k − 1 ◆ q (37) Proof. Consider k ⇥ n matrices in reduced echelon [28]. If the leading 1 in the last row is in the last column, then the other entries in the last row and column are zero. Deleting the other entries produces a (k − 1) ⇥ (n − 1) matrix in reduced echelon. Otherwise, the last column is arbitrary which implies there are qk possibilities for it and deleting it leaves a k ⇥ (n − 1) matrix in reduced echelon [12]. Lemma 16 (Recursive Property of the qt-Binomial Coecients). X μ?n qn(μ0)t−n(μ) ✓ !i μ ◆ qt = X ⌧?n qn(⌧0)t−n(⌧) ✓ ! ⌧ ◆ + X ⌫?(n−1) qn(⌫0)+#it−n(⌫0)+#it−n()+1−i ✓ ! ⌫ ◆ qt (38) Proof. While a proof of this property does exist, it is algebraic in nature rather than com-binatorial and so is not included in this list [16]. The symmetric property of the binomial coecients states that terms in a sequence are symmetric. This property is also easily demonstrable in Pascal’s Triangle. Note that the triangle is symmetric across a vertical line drawn from the tip of the Pascal’s Triangle to the base. 21 Lemma 17 (Symmetric Property for the Classic Binomial Coecients). ✓ n k ◆ = ✓ n n k ◆ (39) Proof. The combinatorial proof is short, but clear. Every k-element subset is determined by the k elements it contains and also by the n k elements it does not contain [25] . Lemma 18 (Symmetric Property for the q-Binomial Coecients). ✓ n k ◆ q = ✓ n n k ◆ q (40) Proof. The proof arises from the fact that there is a bijection between subspaces of dimension k of a vector space and the subspaces of codimension k of the dual space [13]. Lemma 19 (Symmetric Property for the qt-Binomial Coecients). X μ?n ✓ ! μ ◆ = X ⌧?( " −n) ✓ ! ⌧ ◆ qt (41) Proof. While a proof of this property does exist, it is algebraic in nature rather than com-binatorial and so is not included in this list [16]. Lemma 20 (Special Valuations of the Classic Binomial Coecients). ✓ n 0 ◆ = ✓ n n ◆ = 1 (42) ✓ n 1 ◆ = n (43) Proof. Property (42) is fairly obvious. The only subset possible of any n-set is the empty set. Likewise, the only subset of size n which can be formed from the n-set is the n-set itself. Notice also that this property is an extension of the symmetric property, because $n n % = $ n n−n % = $n 0 % [33]. 22 Property (43) is also fairly obvious. The only way in which subsets of size one can be formed is by allowing each element in the original n-set to be its own subset. Lemma 21 (Special Valuations of the q-Binomial Coecients). ✓ n 0 ◆ q = ✓ n n ◆ q = 1 (44) ✓ n 1 ◆ q = 1 qn 1 q = Xn1 i=0 qi = [n]q (45) Proof. This proof follows directly from the symmetry property similarly to the classic case. Lemma 22 (Special Valuations of the qt-Binomial Coecients). ✓ ! 0k ◆ qt = ✓ ! ! ◆ qt = 1 (46) where 0n is the n-part partition whose parts are all zero. Proof. While a proof of this property does exist, it is algebraic in nature rather than com-binatorial and so is not included in this list [16]. Lemma 23 (Classic Binomial Coecients). 2nm ✓ n m ◆ = Xn k=m ✓ n k ◆✓ k m ◆ (47) Lemma 24 (q-Binomial Coecients). qμ2 ⌫Y1 k=μ (1 + qk) ✓ ⌫ μ ◆ = X" μ=⌫ q(2) ✓ ⌫ ! ◆✓ ! μ ◆ (48) Proof. Using the qt-analogue of the property as given below, the q-analogue of the property can be derived algebraically by allowing t = 1. 23 Lemma 25 (qt-Binomial Coecients). t−n(μ)qn(μ0) (1)⌫ (1)μ ✓ ⌫ μ ◆ qt = X μ✓"✓⌫ t−n(")qn("0) ✓ ⌫ " ◆ qt ✓ " μ ◆ qt (49) Proof. An algebraic proof is defined in [16] but is algebraic in nature, so it is not discussed in this paper. Lemma 26 (Special Case of Lemma 23). 2n−1n = Xn k=1 k ✓ n k ◆ (50) Proof. To prove this property it necessary to prove that both sides of the equality count the number of ways in which, if any one element is set aside in its own subset, the number of remaining subsets of any size can be formed. For the left-hand side of the equality, after selecting the element to be isolated, there are 2n−1 ways in which the n1 remaining elements can be formed. Note that this follows the same logic as the proof of Pn k=0 %n k & = 2n, in that each of the n 1 elements can have exactly one out of two possible states: inclusion in the subset or exclusion from the subset. Further, because any of the n elements can be isolated into its own group, the left-hand side, 2n−1n is received. To begin the investigation of the right-hand side, observe that exist %n k & ways of forming a subset with k elements. Further of any of these subsets of k elements, there exist k number of ways in which to choose the element to be isolated. Thus, the left- and right-hand side of the property count the same concept, proving the property [4]. Lemma 27 (Special Case of Lemma (24)). q(μ 2) ⌫Y−1 k=μ (1 qk) X⌫−1 i=0 qi = ✓ ⌫ " ◆ q X"−1 i=0 qi X μ"⌫ q(2) (51) Proof. Using the qt-analogue of the property as given below, the q-analogue of the property can be derived algebraically by allowing t = 1. 24 Lemma 28 (Special Case of Lemma (25)). t−n(μ)qn(μ0) (1)⌫ (1)μ ( Xn i=1 (1 q⌫i)t1−i 1 q )) = X e1✓"✓⌫ t−n(")qn("0)( Xn i=1 (1 q"i)t1−i 1 q ) ✓ ⌫ " ◆ qt (52) Proof. A proof exists of this property in the qt-analogue; however, it is algebraic in nature and so is not included in this paper [16]. 25 Chapter 3 THE PRINCIPAL THEOREM For this study the combinatorial formula, we first rewrite the algebraic formula given by Coskun [16] as a single sum in two dimensions. Theorem 4 (The Principal Theorem). For z = (!1,!2) and μ = (μ1, μ2) ✓ ! μ ◆ = q!1μ1−μ1 2 (μ1−1)+!2μ2−μ2 2 (μ2−1)tμ2(−1) μ (q−!1t−1)μ2(q−!2)μ2 (q)μ2(qμ1−μ2t)μ2(q)2 μ1−μ2(t)μ1−μ2 · Xμ1 i=μ2 ti+μ2q−(i−μ2)(!1−!2)(t)i−μ2(qμ1−i+1)i−μ2(qμ2−!2)i−μ2 · (qi−μ2+1)μ1−i(t)μ1−i(q−!1+i)μ1−i % (53) Proof. As stated previously, the above Lemma is derived from Formula (25). We prove this result in seven steps below: ✓ ! μ ◆ q,t = q μ t2n(μ)+(1−n) μ (qtn−1)μ Y 1i j n, is reduced to 1 i < j 2 which implies i = 1 and j = 2. Furthermore, the factor, (qtn−1)μ, reduces as so: (qtn−1)μ = Yn i=1 (qtn−i)μi = Y2 i=1 (qt2−i)μi = (qt)μi(q)μj (55) For the purposes of generality, the indices, i and j are left as is; they are replaced with their indicated values later in the proof as is n = 2. Thus, the following formula is received: ✓ ! μ ◆ = f!μ(q, t)!!/μ(q!1t, q!2 ; q, t) (56) 26 where f!μ(q, t) = q μ t2n(μ0)+(1−n) μ (qtj−i)μi−μj (qt)μi(q)μj (qtj−i−1)μi−μj (57) and where !!/μ(y, z; q, t) is the recurrence relation for the skew partition "/μ which leaves a horizontal strip defined as, "1 ! μ1 ! "2 ! μ2 . . . μn ! "n+1 = μn+1 = 0 for " = ("1"2, . . . "n) and μ = (μ1, μ2, . . . ,μn) so that !!/μ(y, z; q, t) = X ⌫"! !!/⌫(yt−l; q, t)!!/μ(z; q, t) (58) where ⌫ is all such partitions within μ which create a horizontal strip and whose second row is 0, i.e., the set of all ⌫ is the set (μ2, 0), (μ2 + 1, 0), . . . , (μ1, 0). By substituting Formula (58) into Formula (56) the following formula is received: ✓ " μ ◆ = f!μ(q, t) X ⌫ t μ − ⌫ !μ/⌫(q!1 ; q, t)!⌫(q!2 ; q, t) (59) where !!/μ(x; q, t) = (−q/x)− ! + μ q−n(!0)+n(μ0)H!/μ(q, t) (x−1)! (x−1)μ (60) and H!/μ(q, p, t, b) = Y⇢ (qμi−μj−1tj−i)μj−1−!j (q!i+!j t3−j−ib)μj−1−!j (qμi−μj−1+1tj−i−1)μj−1−!j (q!i+!j+1t2−j−ib)μj−1−!j & · Y(q!i−μj−1+1tj−i−1)μj−1−!j (q!i−μj−1+1tj−i)μj−1−!j Y 1<(j−1)n (qμi+!j+1t1−j−ib)μj−1−!j qμi+!j t2−j−ib)μj−1−!j (61) so that !μ/⌫ = (−q1−!1)− μ + ⌫ q−n(μ0)+n(⌫0)H!/μ(q, t) (q−!1)μ (q−!1)μ (62) !⌫ = (−q1−!2)− ⌫ q−n(⌫0)H⌫(q, t)(q−!2)⌫ (63) 27 where Hμ/⌫(q, t) = Y 1i>< >>: Q2 k=1 (t1−kq−"1)μk(q−"2)vk Q2 k=1 (t1−kq−"1)vk 9>>= >>; (67) Now substitute Formula (64) into Formula (67). Additionally group all non-q-Pochhammer terms into g"μ(q, t, ⌫) so that: g"μ(q, t, ⌫) = t μ −⌫1(1)− μ +2⌫1q− μ −"1 μ −⌫1"1+⌫1"2−(μ1 2 )−(μ2 2 ) (68) Thus, the following definition is received: ✓ ! μ ◆ = f"μ(q, t) X ⌫ g"μ(q, t) (t)⌫1−μ2(qμ1−⌫1+1)⌫1−μ2 (q)⌫1−μ2(qμ1−⌫1t)⌫i−μ2 · Q2 k=1 (q−"1t1−k)μk Q2 k=1 (q−"2)⌫k Q2 k=1 (t1−kq−"1+1)⌫k (69) 28 For the next step it is realized that, because ⌫2 = 0 the following can be reduced as so: Y2 k=1 (t1kq!2)⌫k = (q!2)⌫1 (70) Consequently, after some cancellations of the q-Pochhammer factors inside the summation, the following is obtained: ✓ " μ ◆ = f!μ(q, t) (q)μ1μ2(t)μ1μ2 X ⌫ g!μ(q, t)(t)⌫1μ2(qμ1⌫1+1)⌫1μ2(q!2)⌫1 · (q⌫1μ2+1)μ1⌫1 Q2 k=1 (t1kq!1)μk Q2 k=1 (t1kq!1)⌫k (71) In the next step, after some cancellations amongst the q-Pochhammer factors, all factors independent of ⌫ are taken outside of the summation. ✓ " μ ◆ = f!μ(q, t) Q2 k=1 (t1kq!1)μk (q)μ1μ2(t)μ1μ2(q!1)μ1 X ⌫ g!μ(q, t)(t)⌫1μ2(qμ1⌫1+1)⌫1μ2(q!2)⌫1 · (q⌫1μ2+1)μ1⌫1(t)μ1⌫1(q⌫1!1)μ1⌫1 (72) Finally, the g!μ(q, t, ⌫) factors independent of ⌫ are also taken outside of the summation and, because each ⌫ is a partition such that ⌫2 = 0 μ2 ⌫1 μ1 the formula can be written as a single sum like so which is the given lemma: ✓ " μ ◆ = q!1μ1μ1 2 (μ11)+!2μ2μ2 2 (μ21)tμ2(−1) μ (q!1t1)μ2(q!2)μ2 (q)μ2(qμ1μ2t)μ2(q)2 μ1μ2(t)μ1μ2 · Xμ1 i=μ2 [ti+μ2q(iμ2)(!1!2)(t)iμ2(qμ1i+1)iμ2(qμ2!2)iμ2 · (qiμ2+1)μ1i(t)μ1i(q!1+i)μ1i] (73) 29 Chapter 4 A NEW COMBINATORIAL INTERPRETATION IN TWO DIMENSIONS Upon simplification to a single sum in two dimensions, it was made possible for a new combinatorial interpretation to be made in terms of lattice paths by examining each separate factor in the formula above. The combinatorial interpretation presented in this paper relies on the Young diagrams created for two partitions, ! = (!1,!2) and μ = (μ1, μ2) where μ ✓ !. (Note that μ ✓ ! $ μi !i, 8i.) The Durfee rectangle of μ is defined as the largest rectangular partition which fits inside the Young diagram of μ. The lattice paths upon which the combinatorial interpretation presented in this paper relies are all those which begin at the bottom left corner of the Young diagrams, wrap around the Durfee rectangle of μ or each square extending from the Durfee rectangle of μ, and then extend towards the end of !. Furthermore, the extension of μ is defined as the set of all squares in the first row of μ1 not in the Durfee rectangle. Likewise, the extension of ! is the set of all squares in the first row of ! not in the Durfee rectangle of !. Thus, the definition as given in Formula (53) combinatorially gives way to the following corollary. Corollary 1. For ! = (!1,!2) and μ = (μ1, μ2) ✓ z μ ◆ = q!(",μ)tμ2(−1) μ (q"1t1)μ2(q"2)μ2 (q)μ2(q#(μ)+1t)μ2(q)2 #(μ)(t)#(μ) · X L [t⇢(μ)q⇢(μ)#(")(t)⇢(μ)(q%(μ)+1)⇢(μ)(qμ2"2)⇢(μ) · (q⇢(μ)+1)%(μ)(t)%(μ)(q%("))%(μ)] (74) where "(μ) is the number of squares in the extension of μ, ⇢(μ) is the number above the current lattice path, $(μ) is the number of squares below the lattice path, μ2 is the number of squares in the second row of the Young diagram of μ, !2 is the number of squares in the 30 second row of !, μ is the total number of squares in μ, and the sum is over all lattice paths as described above. Note that "(!, μ) = q1μ1μ1 2 (μ11)+2μ2μ2 2 (μ21) is defined in terms of the Young diagram as well; however, computing "(!, μ) is lengthy, so it is discussed below in its own section. Proof. Note that in Formula (53), certain di↵erences are common occurrences. Each of these di↵erences have a combinatorial meaning which is now explained. μ1 μ2 is the di↵erence between mu1 and μ2, i.e., the number of squares in the extension of μ. This value is assigned the notation #(μ). i μ2 is the di↵erence between the ith square in the extension of μ and the second row of μ. This value is assigned the notation ⇢(μ). μ1i is the di↵erence between the first row of μ and the number of squares up to and including the current square. This value is assigned the notation %(μ). μ is the sum of all rows in the Young tableau. Figure 4: All lattice paths for ! = (10, 8) and μ = (5, 2) All factors in the corollary are split either into the external factors group or the internal factors group based on their placement in the corollary. The external factors are the factors which are placed outside of the summation, and the internal factors are the factors which are placed within the summation. The internal factors are further divided into the ↵ and ' groups based on which squares’ weights in the Young diagram the factors adopt. Each of these groups are discussed separately in the following subsections. 31 q-Pochhammer External Factors The external factors are the factors which exist outside of the summation. A diagram of each factor and its weight distribution is provided in Figure (5). Figure 5: The external non-q-Pochammer factors and their weight distributions for = (10, 8) and μ = (5, 2).. 0.0.3 Non q-Pochhammer External Factors The non-q-Pochhammer external factors consist of the following factors: {q!(",μ), tμ2 , (1) μ } (75) where q!(",μ) = q"1μ1−μ1 2 (μ1−1)+"2μ2−μ2 2 (μ2−1). It is determined that each of these factors (where, using properties of exponents, q!(",μ) is further split into a product) can be defined in terms of the weights w of q and the weights w0 of t. 32 Lemma 29 (Non q-Pochhammer External Factors). Each of the factors placed externally of the summation and which are not q-Pochhammer can be defined in terms of the weights w and w0. q1μ1 = Y w2s qw (76) q2μ2 = Y w2s qw (77) q μ1(μ11) 2 = Y w2s qw/2 (78) q μ2(μ21) 2 = Y w2s qw/2 (79) tμ2 = Y w2s tw (80) (1) μ = Y w2s (1)w (81) Proof. The following is a description for each factor and the distribution of its weights in the Young diagram of and μ. First Non q-Pochhammer External Factor q1μ1 = Y w2s qw (82) where s is each square in the first row of μ and w is the number of squares in the first row of . Second Non q-Pochhammer External Factor q2μ2 = Y w2s qw (83) where s is each square in the first row of μ in the Durfee rectangle and w is the weight in each s where w is the number of squares in the second row of . 33 Third Non q-Pochhammer External Factor q μ1(μ11) 2 = Y w2s qw/2 (84) where s is each square in the first row of μ and w is the weight in each s where w is one less than the number of squares in the first row of μ. Fourth Non q-Pochhammer External Factor q μ2(μ21) 2 = Y w2s qw/2 (85) where s is each square in the first row of μ in the Durfee rectangle and w is the weight in each s where w is one less than the number of squares in the second row of μ. Fifth Non q-Pochhammer External Factor tμ2 = Y w2s tw (86) where s is each square in the first row of μ in the Durfee rectangle and w is the weight of each s such that w is one. Sixth Non q-Pochhammer External Factor (1) μ = Y w2s (1)w (87) where s is all squares in the extension of μ and w is 1. 0.0.4 q-Pochhammer External Factors The q-Pochhammer external factors consist of the following factors: {(q−!1t−1)μ2 , (q−!2)μ2 , (q)−1 μ2 , (q"(μ)+1t)−1 μ2 , (q)−2 "(μ), (t)−1 "(μ)} (88) 34 The q-Pochhammer external factors all assign to each square, s, in the first row of μ in the Durfee rectangle the weights w and w0. The following lemma is a list of each factor and its definition in terms of its weights w and w0. Lemma 30. Each factor is defined in terms of a certain weight distribution unique to that factor. (q−!1t−1)μ2 = Q s (1 qwtw0) q⌫(w)t⌫(w0) (89) (q−!2)μ2 = Y s (1 q−wtw0) (90) (q)−1 μ2 = Y s (1 qwtw0) (91) (q#(μ)+1t)−1 μ2 = Y s (1 qwtw0) (92) (q)−2 #(μ) = Y s (1 qwtw0)2 (93) (t)−1 #(μ) = Y s (1 qwtw0) (94) A diagram of each q-Pochhammer external factor and their weight distributions is pro-vided in Figure (6). Proof. The weight distribution of each q-Pochhammer external factor is described below. First q-Pochhammer External Factor (q−!1t−1)μ2 = Q s (1 qwtw0) q⌫(w)t⌫(w0) (95) where ⌫(w) = P s w, s is each square in the first row of the Durfee rectangle of μ, w is one more than the number of squares to the right of s, and w0 is 1. 35 Figure 6: The external q-Pochammer factors and their weight distributions for = (10, 8) and μ = (5, 2). Second q-Pochhammer External Factor (q−!2)μ2 = Y s (1 q−wtw0) (96) where s is each square in the first row of the Durfee rectangle of μ, w is one more than the number of squares to the right of s not in the extension of and w0 is 0. Third q-Pochhammer External Factor (q)−1 μ2 = Y s (1 qwtw0) (97) where s is each square in the first row of the Durfee rectangle of μ, w is one more than the number of squares to the right of s in the Durfee rectangle and w0 is 0. Fourth q-Pochhammer External Factor (q"(μ)+1t)−1 μ2 = Y s (1 qwtw0) (98) 36 where s is each square in the first row of the Durfee rectangle of μ, w is one more than the number of squares to the right of s and w0 is 1. Fifth q-Pochhammer External Factor (q)−2 !(μ) = Y s (1 qwtw0)2 (99) where s is each square in the extension of μ, w is one more than the number of squares to the left of s in the extension of μ, and w0 is 0. Sixth q-Pochhammer External Factor (t)−1 !(μ) = Y s (1 qwtw0) (100) where s is each square in the extension of μ, w is the number of squares to the right of s in the extension of μ, and w0 is 1. Internal Factors The internal factors are those factors which are internal to the summation: {t−⇢(μ), q−⇢(μ)!(#), (t)⇢(μ), (q$(μ)+1)⇢(μ), (qμ2−#2)⇢(μ), (q⇢(μ)+1)$(μ), (t)$(μ), (q−$(#))$(μ)} (101) The internal factors are then further broken down into the ↵ factors group and the beta factors group. The ↵ and " factors both assign weights w and w0 to each square, s, in the extension of μ. Because the internal factors are internal to the summation which runs over all lattice paths, each lattice path will impact which squares’ weights the factors will assume. The ↵ factors will assume the weights of the squares above the current lattice path while the " factors assume the weights of the squares below the current lattice path. 37 0.0.5 ↵ Factors The alpha internal factors are the following: {(t)⇢(μ), (q"(μ)+1)⇢(μ), (qμ2−#2)⇢(μ), (qμ2−#2)⇢(μ)} (102) As stated previously, the ↵ factors assign weights w and w0 to each square, s, in the extension of μ and assume the weights of the squares above the current lattice path. The definition of each factor in terms of its weights w and w0 is listed below in the following lemma. Lemma 31. Each internal factor in the ↵ group can be defined in terms of a weight distri-bution unique to that factor. q−⇢(μ)$(#)t−⇢(μ) = 1 qwtw0 (103) (t)⇢(μ) = Y s (1 qwtw0) (104) (q"(μ)+1)⇢(μ) = Y s (1 qwtw0) (105) (qμ2−#2)⇢(μ) = Y s (1 q−wtw0) (106) Proof. The following is a list of each factor, its definition in terms of its weight distribution, and a description of how its weight is distributed. Non q-Pochhammer Factor q−⇢(μ)$(#)t−⇢(μ) = 1 qwtw0 (107) where w is the number of squares in the extension of " and w0 is 1. An example is provided in Figure (7). 38 Figure 7: t−⇢(μ)q−⇢(μ)"(#) for = (10, 8) and μ = (5, 2) First q-Pochhammer Factor (t)⇢(μ) = Y s (1 qwtw0) (108) where w is the number of squares to the left of s in the extension of μ and w0 is 1. An example is provided in Figure (8). Figure 8: (t)⇢(μ) for = (10, 8) and μ = (5, 2) Second q-Pochhammer Factor (q$(μ)+1)⇢(μ) = Y s (1 qwtw0) (109) where w is one more than the number of squares to the right of s in the extension of μ and w0 is 0. An example is provided in Figure (9). 39 Figure 9: (q!(μ)+1)⇢(μ) for = (10, 8) and μ = (5, 2) Third q-Pochhammer Factor (qμ2−#2)⇢(μ) = Y s (1 q−wtw0) (110) where w is the number of squares to the right of s that are not in the extension and w0 is 0. An example is provided in Figure (10). Figure 10: (qμ2−#2)⇢(μ) for = (10, 8) and μ = (5, 2) 40 0.0.6 Group The group of internal factors are the following: {(q⇢(μ)+1)"(μ), (t)"(μ), (q−"(#))"(μ)} (111) As stated above, the factors assign weights w and w0 to each square, s, in the extension of μ and assume the weights of the squares below the current lattice path. Each factor within the group of internal factors is then defined in terms of these weight distributions as shown in the below lemma. Lemma 32. Each factor within the group of internal factors can be defined in terms of the weight distribution unique to that factor. (q⇢(μ)+1)"(μ) = Y s (1 qwtw0) (112) (t)"(μ) = Y s (1 qwtw0) (113) (q−"(#))"(μ) = Y s (1 q−wtw0) (114) First q-Pochhammer Factor (q⇢(μ)+1)"(μ) = Y s (1 qwtw0) (115) where w is one more than the number of squares to the left of s in the extension of μ and w0 is 0. An example is provided in Figure (11). 41 Figure 11: (q⇢(μ)+1)"(μ) for = (10, 8) and μ = (5, 2) Second q-Pochhammer Factor (t)"(μ) = Y s (1 qwtw0) (116) where w is the number of squares to the right of s in the extension of μ and w0 is 1. An example is provided in Figure (12). Figure 12: (t)"(μ) for = (10, 8) and μ = (5, 2) 42 Third q-Pochhammer Factor (q−!("))!(μ) = Y s (1 q−wtw0) (117) where w is one more than the number of squares to the right of s and w0 is 0. An example is provided in Figure (13). Figure 13: (q−!("))!(μ) for ! = (10, 8) and μ = (5, 2) A Complete Example Using the new combinatorial interpretation presented in this paper, a complete demon-stration is performed for the reader in this section with a complete description of each external and internal factor and their weight distributions for the partitions ! = (4, 2) and μ = (3, 1). The results of the product of the external non-q-Pochhammer factors, the exter-nal q-Pochhammer factors, the product of the internal ↵ and # factors corresponding to each lattice lath, and the sum of the internal factors over all lattice paths are separately discussed. Finally, the product of each of these separate results is calculated and then compared to both the Okounkov combinatorial interpretation and the Coskun definition. 0.0.7 External Non q-Pochhammer Factor Diagrams of the external non-q-Pochhammer factors are provided in Figure (14). 43 Figure 14: The external non-q-Pochhammer factors for = (4, 2) and μ = (3, 1) Each square in the first row of the Durfee rectangle of μ is assigned the weight w = 2 so that q1μ1 = q12 (118) Each square in the first row of the Durfee rectangle of μ is assigned the weight w = 2 so that q2μ2 = q2 (119) Each square in the first row of μ is assigned the weight w = 2 so that: q μ1(μ11) 2 = q3 (120) Each square in the first row of the Durfee rectangle of μ is assigned the weight w = 0 so that: q μ2(μ21) 2 = q0 = 1 (121) 44 Each square in the first row of the Durfee rectangle of μ is assigned the weight w0 = 1 so that: tμ2 = t1 = t (122) Each square in the extension of μ is assigned the weight w = 1 so that: (1) μ = (1)2 = 1 (123) Multiplying all the end results of the above together outputs q11t as the product of the all the external non-q-Pochhammer factors. 0.0.8 External q-Pochhammer Factor A diagram displaying each factor and its weight distribution is provided in Figure (15) Figure 15: The external q-Pochhammer factors for = (4, 2) and μ = (3, 1) Each square in the first row of the Durfee rectangle of μ is assigned the weights w = {4} and w0 = {1} respectively so that (q−1t−1)μ2 = 1q−4t−1 (124) 45 Each square in the first row of the Durfee rectangle of μ is assigned the weights w = {2} and w0 = {0} respectively so that (q−!2)μ2 = 1q−2 (125) Each square in the first row of the Durfee rectangle of μ is assigned the weights w = {1} and w0 = {0} respectively so that (q)−1 μ2 = 1q (126) Each square in the first row of the Durfee rectangle of μ is assigned the weights w = {3} and w0 = {1} respectively so that (q"(μ)+1t)−1 μ2 = 1 (1 q3t) (127) Each square in the extension of the Durfee rectangle of μ is assigned the weights w = {1, 2} and w0 = {0} respectively so that (q)"(μ)2 = 1 (1 q)2(1 q2)2 (128) Each square in the extension of the Durfee rectangle of μ is assigned the weights w = {1, 0} and w0 = {1} respectively so that (t)−1 "(μ) = 1 (1 t)(1 qt) (129) Multiplying all of these external q-Pochhammer factors results in (1 q−2)(1 q−4) (1 q)3(1 q2)2(1 t)(1 qt)(1 q3t) . (130) 46 0.0.9 Internal Non q-Pochhammer Factor Because the summation is taken over all lattice paths, the internal factors will be grouped by lattice path and then added at the end of this section. However, the weight distributions are discussed here, because no matter the lattice path the weight distribution will be the same for each factor for all lattice paths. Also note that the weights for the internal factors are always assigned to the squares in the extension of μ. • For t−⇢(μ)q−⇢(μ)"(#) each square is assigned the weights w = {2} and w0 = {1} respec-tively. • For (t)⇢(μ) each square is assigned the weights w = {0, 1} and w0 = {1} respectively. • For (q$(μ)+1)⇢(μ) each square is assigned the weights w = {2, 1} and w0 = {1} respec-tively. • For (qμ2−#2)⇢(μ) each square is assigned the weights w = {1, 0} and w0 = {0} respec-tively. • For (q⇢(μ)+1)$(μ) each square is assigned the weights w = {1, 2} and w0 = {0} respec-tively. • For (t)$(μ) each square is assigned the weights w = {2, 1} and w0 = {1} respectively. • For (q−$(#))⇢(μ) each square is assigned the weights w = {3, 2} and w0 = {1} respectively. First Lattice Path: A figure demonstrating the first lattice path and the internal factors and their weight distributions is provided in Figure (16). • t−⇢(μ)q−⇢(μ)"(#) = 1, because there are no squares above the first lattice path. • (t)⇢(μ) = 1, because there are no squares above the first lattice path. • (q$(μ)+1)⇢(μ) = 1, because there are no squares above the first lattice path. 47 • (qμ2!2)⇢(μ) = 1, because there are no squares above the first lattice path. • (q⇢(μ)+1)#(μ) = (1 q)(1 q2), because the weights of the two squares beneath the lattice path are adopted. • (t)#(μ) = (1t)(1qt) because the weights of the two squares beneath the lattice path are adopted. • (q#(!))⇢(μ) = (1 q3)(1 q2), because the weights of the two squares beneath the lattice path are adopted. Therefore, the product of the factors resulting from the first lattice path is (1 q3)(1 q2)(1 q)(1 q2)(1 t)(1 qt). Figure 16: The first lattice path for = (4, 2) and μ = (3, 1) 48 Second Lattice Path: A figure demonstrating the second lattice path and the internal factors and their weight distributions is provided in Figure (17). • t⇢(μ)q⇢(μ)"(#) = 1 q2t, because the weight of the one square above the second lattice is adopted. • (t)⇢(μ) = 1t, because the weight of the one square above the second lattice is adopted. • (q$(μ)+1)⇢(μ) = 1, because the weight of the one square above the second lattice is adopted. • (qμ2#2)⇢(μ) = 1, because the weight of the one square above the second lattice is adopted. • (q⇢(μ)+1)$(μ) = (1q)(1q2), because the weight of the one square beneath the lattice path is adopted. • (t)$(μ) = (1 t)(1 qt) because the weight of the one square beneath the lattice path is adopted. • (q$(#))⇢(μ) = (1 q3)(1 q2), because the weight of the one square beneath the lattice path is adopted. Therefore, the product of the factors resulting from the second lattice path is (1q2)(1q1)(1q2)2(1t)2 q2t . Third Lattice Path: A figure demonstrating the third lattice path and the internal factors and their weight distributions is provided in Figure (18). • t⇢(μ)q⇢(μ)"(#) = 1 q4t2, because the weight of both squares above the second lattice are adopted. • (t)⇢(μ) = (1 t)(1 qt), because the weight of both squares above the second lattice are adopted. 49 Figure 17: The second lattice path for = (4, 2) and μ = (3, 1) • (q!(μ)+1)⇢(μ) = (1 q)(1 q2), because the weight of both squares above the second lattice are adopted. • (qμ2#2)⇢(μ) = 0, because the weight of both squares above the second lattice are adopted. • (q⇢(μ)+1)!(μ) = 1, because there are no more squares below the lattice path. • (t)!(μ) = 1 because there are no more squares below the lattice path. • (q!(#))⇢(μ) = 1, because there are no more squares below the lattice path. Therefore, the product of the factors resulting from the third lattice path is 0. 50 Figure 18: The third lattice path for = (4, 2) and μ = (3, 1) To finish with the internal factors, simply take their sum and simplify to receive (1 + q)4(1 + q)2(1 + t)(1 q q2t + qt2 + q2t2 + q3t2) q5t . (131) Finally, to complete the computation of the qt-binomial coecients using the new combi-natorial interpretation simply find and simplify the product of the external non-q-Pochhammer factors, the external q-Pochammer factors, and the sum of the internal factors to receive: ✓ μ ◆ = (1 + q)(1 + q4t)(1 q q2t + qt2 + q2t2 + q3t2) t(1 + qt)(1 + q3t) (132) which is the exact same answer as those found with Definitions (25) and (8). Additionally, the answer was computed in 0.009 seconds–the fastest time of all three methods. 51 Chapter 5 FUTURE RESEARCH As stated in the introduction, this new combinatorial interpretation of the qt-binomial coecients theoretically makes the qt-binomial coecients more comprehensible and greatly simplifies the computation of the qt-binomial coecients; however, a dilemma rests in the fact that this combinatorial interpretation is only in terms of the second dimension. In order for this interpretation to reach its fullest potential, it must be generalized to the nth-dimension. This can feasibly be done following the same methods as the author using algebraic manip-ulation and experimentation via Wolfram Mathematica (or any other programming tool). Once this is done, combinatorial proofs of the properties of the qt-binomial coecients as listed in Section 2.4 may be devised. This combinatorial interpretation can also have further massive impact in the field of combinatorics. Many classic cases of special sequences of numbers (such as Stirling, Lah, Bell, Bernoulli, Fibonacci, Catalan, etc.) are defined in terms of the classic binomial coecients [26]. Similarly, their q- and qt-analogues are defined in terms of the q- and qt-binomial coecients. Therefore, it is expected that using this new combinatorial interpretation as a foundation, the qt-analogues of many of these special numbers will find a new combinatorial interpretation as well. Lastly, as is typical of many topics of study in combinatorics, it is believed that this combinatorial interpretation will find applications in fields of, not only mathematics, but also applied fields such as operations research, computer science, electrical engineering and statics, statistical physics, chemistry, and molecular biology [23] where combinatorial interpretations are necessary to mathematically model real-world situations. Perhaps the generalized nature of the qt-binomial coecients will open the way to its use in new fields as well. 52 REFERENCES [1] G. Andrews, R. Roy, and R. Askey, Special functions., Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 1999. [2] J. Azose, A tiling interpretation of q-binomial coecieints, Ph.D. thesis, Harvey Mudd College, May 2007. [3] E. Bender and S. Williamson, Foundations of applied combinatorics, Addison-Wesley Publishing Company, 1991. [4] A. Benjamin and J. Quinn, Proofs that really count: The art of combinatorial proof, The Dociani Mathematical Expositions, no. 27, The Mathematical Association of America, 200. [5] C. Berge, Principles of combinatorics, Mathematics in Science and Engineering, vol. 72, Academic Press, 1971. [6] G. Berman and K.D. Fryer, Introduction to combinatorics, Academic Press, 1972. [7] N. Biggs, K. Lloyd, and R. Wilson, Handbook of combinatorics, ch. The History of Combinatorics, The MIT Press, 1995. [8] M. B´ona, Combinatorics of permutations, Discrete Mathematics and Its Applications, CRC Press, 2012. [9] R. Brualdi, Introductory combinatorics, North-Holland, 1977. [10] V. Bryant, Aspects of combinatorics: A wide-ranging introduction, Cambridge Univer-sity Press, 1993. [11] D. Burton, The history of mathematics: An introduction, 7th ed. ed., McGraw-Hill, 2011. 53 [12] P. Cameron, Combinatorics: Topics, techniques, algorithms, Cambridge University Press, 1994. [13] , Notes on counting, Citeseer, 2010. [14] S. Cooper, B. Berndt, N. Deka, T. Huber, and M. Schlosser (eds.), Ramanujan redis-covered: Proceedings of a conference on elliptic functions, partitions, and q-series in memory of k. venkatachaliengar, 2009. [15] H. Coskun, Multiple bracket function stirling number, and lah number identities, Con-temporary Mathematics (2000), 1–23. [16] , Multiple analogues of binomial coecients and families of related special num-bers, Discrete Mathematics 310 (2010), no. 17-18, 2280–2298. [17] M. Erickson, Introduction to combinatorics, John Wiley Sons, Inc., 1996. [18] L. Euler, Introductio in analysin infinitorum, auctore leonhardo eulero..., apudMarcum- Michaelem Bousquet, 1748. [19] D. Foata and G. Han, q-series in combinatorics; permutation statistics, Preliminary version (2004). [20] C.F. Gauss, Werke, Leipzig: Teubner 3 (1863). [21] P. Goetgheluck, Computing binomial coecients, The American Mathematical Monthly 94 (1987), no. 4, 360–365. [22] R.L. Graham and L. Grotschel, M.; Lovasz, Handbook of combinatorics, vol. 2, TheMIT Press, 1995. [23] , Handbook of combinatorics, vol. 1, The MIT Press, 1995. [24] E. Heine, Untersuchungen u ber die reihe, J. Reine Angew. Math. 34 (1847), 285–328. 54 [25] B. Kisacanin, Mathematical problems and proofs: Combinatorics, number theory, and geometry, Plenum Press, 1998. [26] J. Konvalina, A unified interpretation of the binomial coecients, the stirling numbers, and the gaussian coecients, The American Mathematical Monthly 107 (2000), no. 10, 901–910. [27] A. Lascoux and S. Warnaar, Branching rules for symmetric functions and sln basic hypergeometric series, Advances in Applied Mathematics 46 (2011), no. 1, 424–456. [28] J.H. Lint and R.M. Wilson, A course in combinatorics, Cambridge University Press, 1992. [29] I.G. Macdonald, Symmetric functions and hall polynomials, 2nd ed. ed., Oxford Uni-versity Press, 1995. [30] R. Merris, Combinatorics, PWS Publishing Company, 1995. [31] A. Okounkov, Binomial formula for macdonald polynomials and its applications, Math-ematical Research Letters 4 (1997), 533–553. [32] A. Okounkov, Combinatorial formula for Macdonald polynomials, Bethe Ansatz, and generic Macdonald polynomials, ArXiv Mathematics e-prints (2000). [33] G. Polya, Lehmer D., M. Phister, J. Riordan, E. Montroll, N.G. Bruijn, F. Harary, R. Bellman, R. Kalaba, E. Peterson, L. Breiman, A. Tucker, E. Beckenbach, M. Hall, J. Wolfowitz, C. Tompkins, K. Trueblood, G. Gamow, and H. Weyl, Applied combina-torial mathematics, John Wiley and Sons, Inc., 1964. [34] A. Prasad, Counting subspaces of a finite vector space—2, Resonance 15 (2010), no. 12, 1074–1083. 55 [35] B.. Sagan and C. Savage, Combinatorial interpretations of binomial coecient ana-logues related to lucas sequences, Integers: Electronic Journal of Combinatorial Number Theory. [36] M. Shattuck, On some relations satisfied by the p; q-binomial coecient, Siauliai Math-ematical Seminar 6 (2011), no. 14, 69–84. [37] R. Stanley, Enumerative combinatorics, vol. 1, Cambridge University Press, December 2011. [38] , Enumerative combinatorics, vol. 2, Cambridge University Press, December 2011. [39] D. Stanton and D. White, Constructive combinatorics, Undergraduate Texts in Math-ematics, Springer-Verlag, 1986. [40] RJ Wilson (ed.), Applications of combinatorics, Shiva Publishing Limited, 1981. [41] A. Yong, What is a young tableau?, Notices of the AMS 54, no. 2. 56 VITA Rachel Landers received two Bachelor of Arts degrees in Mathematics and Spanish, respectively, in December 2012. She graduated cum laude with honors. In May 2015 she will graduate with her Master of Science in Mathematics from Texas A&M University-Commerce. Permanent Address: Texas A&M University-Commerce Department of Mathematics P.O. Box 3011 Commerce, TX 75429 Email: rachel.landers12@gmail.com |