| 2. |
- Rosengren, Hjalmar, 1972, et al.
(författare)
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On a hypergeometric identity of Gelfand, Graev and Retakh
- 2003
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Ingår i: Journal of Computational and Applied Mathematics. ; 160, s. 147-158
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Tidskriftsartikel (refereegranskat)abstract
- A hypergeometric identity equating a triple sum to a single sum, originally found by Gelfand, Graev and Retakh [Russian Math. Surveys 47 (1992), 1-88] by using systems of differential equations, is given hypergeometric proofs. As a bonus, several q-analogues can be derived.
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| 3. |
- Sjöstrand, Jonas, 1981-
(författare)
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Enumerative combinatorics related to partition shapes
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis deals with enumerative combinatorics applied to three different objects related to partition shapes, namely tableaux, restricted words, and Bruhat intervals. The main scientific contributions are the following. Paper I: Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2]. We prove a generalisation of this conjecture using the Robinson-Schensted correspondence and a new concept called chess tableaux. The proof is built on a remarkably simple relation between the sign of a permutation pi and the signs of its RS-corresponding tableaux P and Q, namely sgn(pi) = (−1)^v sgn(P)sgn(Q), where v is the number of disjoint vertical dominoes that fit in the partition shape of P and Q. The sign-imbalance of a partition shape is defined as the sum of the signs of all standard Young tableaux of that shape. As a further application of the sign-transferring formula above, we also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. Paper II: We generalise some of the results in paper I to skew tableaux. More precisely, we examine how the sign property is transferred by the skew Robinson-Schensted correspondence invented by Sagan and Stanley. The result is a surprisingly simple generalisation of the ordinary non-skew formula above. As an application, we find vanishing weighted sums of squares of sign-imbalances, thereby generalising a variant of Stanley’s second conjecture. Paper III: The following special case of a conjecture by Loehr and Warrington was proved by Ekhad, Vatter, and Zeilberger: There are 10^n zero-sum words of length 5n in the alphabet {+3,−2} such that no consecutive subword begins with +3, ends with −2, and sums to −2. We give a simple bijective proof of the conjecture in its original and more general setting where 3 and 2 are replaced by any relatively prime positive integers a and b, 10^n is replaced by ((a+b) choose a)^n, and 5n is replaced by (a+b)n. To do this we reformulate the problem in terms of cylindrical lattice walks which can be interpreted as the south-east border of certain partition shapes. Paper IV: We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id,pi] of the symmetric group correspond to non-capturing rook configurations on a skew Ferrers board. These intervals turn out to be exactly those whose flag manifolds are defined by inclusions, as defined by Gasharov and Reiner. The characterisation connects Poincaré polynomials (rank-generating functions) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups A_n and B_n. The expressions involve q-Stirling numbers of the second kind, and for the group A_n putting q = 1 yields the poly-Bernoulli numbers defined by Kaneko.
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