Elliptic rook and file numbers

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of Garsia and Remmel's q-file numbers for skyline boards. We also provide an elliptic extension of the j-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings has been taken out to keep the length of this paper reasonable

Elliptic rook and file numbers

In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corre- sponding factorization theorems which in the classical case were established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind as well as Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel

An esoteric identity with many parameters and other elliptic extensions of elementary identities

We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can be obtained from telescoping. So we used telescoping in our study to find elliptic extensions of these identities. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case. In addition, we recover some $q$-identities due to Warnaar.Comment: 15 pages, comments welcom

An elliptic extension of the multinomial theorem

We present a multinomial theorem for elliptic commuting variables. This result extends the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a simple elliptic star-triangle relation, ensuring the uniqueness of the normal form coefficients, and, for the recursion of the closed form elliptic multinomial coefficients, the Weierstra{\ss} type $\mathsf A$ elliptic partial fraction decomposition. From our elliptic multinomial theorem we obtain, by convolution, an identity that is equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel--Turaev ${}_{10}V_9$ summation, which in the trigonometric or basic limiting case reduces to Milne's type $\mathsf A$ extension of the Jackson ${}_8\phi_7$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, our derivation of the $\mathsf A_r$ Frenkel--Turaev summation constitutes the first combinatorial proof of that fundamental identity, and, at the same time, of important special cases including the $\mathsf A_r$ Jackson summation.Comment: 14 p

Elliptic Rook and File Numbers

In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corresponding factorization theorems which in the classical case was established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind, and Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel. This is a joint work with Michael Schlosser.Non UBCUnreviewedAuthor affiliation: Universität WienPostdoctora