Some Fine Combinatorics
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1 Some Fine Combinatorics David P. Little October 25, Fall Eastern Section Meeting of the AMS Penn State University
2 Introduction In Basic Hypergeometric Series and Applications, Fine studied the series F(a,b;t : q) = (aq) n t n (bq) n = aq 1 bq t + (1 aq)(1 aq2 ) (1 bq)(1 bq 2 ) t2 + where (z) n = (1 z)(1 zq) (1 zq n 1 ).
3 Introduction In Basic Hypergeometric Series and Applications, Fine studied the series F(a,b;t : q) = (aq) n t n (bq) n = aq 1 bq t + (1 aq)(1 aq2 ) (1 bq)(1 bq 2 ) t2 + where (z) n = (1 z)(1 zq) (1 zq n 1 ). Some results from Chapter 1: Functional equations satisfied by F(a, b; t : q) Rogers-Fine Identity Symmetry result for (1 t)f(a,b;t : q) Specializations
4 Rogers-Fine Identity Theorem (aq) n t n (bq) n = (aq) n (atq/b) n b n t n q n2 (1 atq 2n+1 ) (t) n+1 (bq) n
5 Rogers-Fine Identity Theorem (aq) n t n (bq) n = We begin by making the substitutions (aq) n (atq/b) n b n t n q n2 (1 atq 2n+1 ) (t) n+1 (bq) n a b aq b c t a
6 Rogers-Fine Identity Theorem (aq) n t n (bq) n = We begin by making the substitutions (aq) n (atq/b) n b n t n q n2 (1 atq 2n+1 ) (t) n+1 (bq) n a b aq b c t a resulting in ( b/a) n a n (cq) n = ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n
7 Weighted Tilings A tiling is a (finite) sequence of squares:
8 Weighted Tilings A tiling is a (finite) sequence of squares: The weight of each tile is given by aq i if t is a with i or to its left w(t) = bq i if t is a with i or to its left c if t is a.
9 Weighted Tilings A tiling is a (finite) sequence of squares: The weight of each tile is given by aq i if t is a with i or to its left w(t) = bq i if t is a with i or to its left c if t is a. The weight of a tiling T is given by w T (a,b,c;q) = w(t). t T
10 Weighted Tilings A tiling is a (finite) sequence of squares: The weight of each tile is given by aq i if t is a with i or to its left w(t) = bq i if t is a with i or to its left c if t is a. The weight of a tiling T is given by w T (a,b,c;q) = w(t). t T w T (a,b,c;q) = c aq c bq 2 c c c bq 6 aq 7 c c bq 9 c aq 11 = a 3 b 3 c 8 q 36.
11 Proof of Rogers-Fine, Preliminaries Only considering tilings that do not end with. Tilings can consist of any finite number of tiles G(a,b,c;q) = w T (a,b,c;q) T
12 Proof of Rogers-Fine, Preliminaries Only considering tilings that do not end with. Tilings can consist of any finite number of tiles G(a,b,c;q) = w T (a,b,c;q) T We will show that ( b/a) n a n = G(a,b,c;q) = (cq) n ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n
13 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares.
14 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares.
15 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares.
16 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares.
17 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares.
18 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares.
19 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares. Weight of the ith segment of tiles: (a + bq n i ) j=0 c j q j(n+1 i) = a + bqn i 1 cq n+1 i.
20 Proof Of Rogers-Fine, Part I ( b/a) n a n (cq) n = G(a,b,c;q) Consider all tilings with exactly n black or gray squares. Weight of the ith segment of tiles: (a + bq n i ) j=0 c j q j(n+1 i) = a + bqn i 1 cq n+1 i. Generating function for tilings with exactly n black or gray squares is n i=1 a + bq n i 1 cq n+1 i = ( b/a) na n (cq) n.
21 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
22 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
23 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
24 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
25 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
26 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
27 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
28 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
29 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right.
30 Weighted Center The weighted center of a tiling is the place on the board where the number of gray or white squares strictly to its left is the same as the number of black or gray squares strictly to its right. The degree of a tiling is the number of gray or white squares strictly to the left of its weighted center.
31 Proof Of Rogers-Fine, Part II G(a,b,c;q) = Consider all tilings of degree n. ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n
32 Proof Of Rogers-Fine, Part II G(a,b,c;q) = ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n Consider all tilings of degree n. Place n gray and/or white squares in positions 1 through n. n (c + bq j 1 ) = ( b/c) n c n j=1
33 Proof Of Rogers-Fine, Part II G(a,b,c;q) = ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n Consider all tilings of degree n. Place n gray and/or white squares in positions 1 through n. n (c + bq j 1 ) = ( b/c) n c n j=1 Place n black and/or gray squares in positions n + 1 through 2n. n (aq n + bq 2n j ) = ( b/a) n a n q n2 j=1
34 Proof Of Rogers-Fine, Part II G(a,b,c;q) = ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n Consider all tilings of degree n. Place n gray and/or white squares in positions 1 through n. n (c + bq j 1 ) = ( b/c) n c n j=1 Place n black and/or gray squares in positions n + 1 through 2n. n (aq n + bq 2n j ) = ( b/a) n a n q n2 j=1 Weighted center may or may not coincide with a gray square. (1 + bq 2n )
35 Proof Of Rogers-Fine, Part II G(a,b,c;q) = Insert black squares before weighted center ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n 1 1 a 1 1 aq 1 1 aq n = 1 (a) n+1
36 Proof Of Rogers-Fine, Part II G(a,b,c;q) = Insert black squares before weighted center ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n 1 1 a 1 1 aq 1 1 aq n = 1 (a) n+1 Insert white squares after weighted center 1 1 cq 1 1 cq cq n = 1 (cq) n
37 Proof Of Rogers-Fine, Part II G(a,b,c;q) = Insert black squares before weighted center ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n 1 1 a 1 1 aq 1 1 aq n = 1 (a) n+1 Insert white squares after weighted center 1 1 cq 1 1 cq cq n = 1 (cq) n Generating function for all tilings of degree n is ( b/a) n ( b/c) n a n c n q n2 (1 + bq 2n ) (a) n+1 (cq) n.
38 Functional Equations Each of the following functional equations follow from the observation that if T has n black squares. w T (aq,b,c;q) = q n w T (a,b,c;q)
39 Functional Equations Each of the following functional equations follow from the observation that if T has n black squares. w T (aq,b,c;q) = q n w T (a,b,c;q) G(a,b,c;q) = 1 + a + b 1 cq G(a,bq,cq;q)
40 Functional Equations Each of the following functional equations follow from the observation that if T has n black squares. w T (aq,b,c;q) = q n w T (a,b,c;q) G(a,b,c;q) = 1 + a + b 1 cq G(a,bq,cq;q) G(a,b,c;q) = 1 c 1 a + b + c 1 a G(aq,bq,c;q)
41 Functional Equations Each of the following functional equations follow from the observation that if T has n black squares. w T (aq,b,c;q) = q n w T (a,b,c;q) G(a,b,c;q) = 1 + a + b 1 cq G(a,bq,cq;q) G(a,b,c;q) = 1 c 1 a + b + c 1 a G(aq,bq,c;q) G(a,b,c;q) = 1 + b (a + b)(b + c)q + 1 a (1 a)(1 cq) G(aq,bq2,cq;q)
42 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i i=0
43 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. i=0
44 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. i=0
45 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. i=0
46 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. i=0
47 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. i=0
48 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. i=0
49 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. i=0
50 Specialization: c = 1 q-analog of binomial series ( b/a) n a n 1 + bq i = (q) n 1 aq i Think of infinitely long sequences, finite number of black and gray squares. Weight of the ith segment of tiles: (1 + bq i ) j=0 i=0 a j q ij = 1 + bqi 1 aq i
51 Symmetry The series (1 a)g(a,b,c;q) is symmetric in the variables a and c. (1 a)g(a,b,c;q) = (1 c)g(c,b,a;q)
52 Symmetry The series (1 a)g(a,b,c;q) is symmetric in the variables a and c. (1 a)g(a,b,c;q) = (1 c)g(c,b,a;q)
53 Symmetry The series (1 a)g(a,b,c;q) is symmetric in the variables a and c. (1 a)g(a,b,c;q) = (1 c)g(c,b,a;q) Reverse order of tiles, convert black squares into white squares and vice versa.
54 Other Results Using Similar Techniques Numerous classical partition identities Lebesgue identities Rogers-Fine q-binomial series Eight (plus five) q-series identities of Rogers q-series symmetry results Future results?
55 q-analog of Gauss s Theorem Weight tiles in the following manner: aq i if t is a in position i cq i if t is a in position i abq i if t is a with i or to its left w(t) = bcq i if t is a with i or to its left cq i if t is a in position i 1 if t is a Consider tilings where each circle is followed by a white tile. Theorem (cq) ( c/a) n ( q/b) n a n b n (q) n (cq) n = n=1 (1 + bcq n 1 )(1 + aq n ) (1 abq n 1 )
56 q-analog of Kummer s Theorem Weight tiles in the following manner: q i if t is a in position i cq i if t is a in position i bq i if t is a with i or to its left w(t) = bcq i if t is a with i or to its left bcq i if t is a in position i if t is a Consider tilings where each circle is followed by a white tile. Theorem (bc) ( c) n ( q/b) n b n (q) n (bc) n = n=1 (1 + q n )(1 + cq 2n 1 )(1 + cb 2 q 2n 2 ) 1 bq n 1
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