116 research outputs found
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
A remark on the enumeration of rooted labeled trees
Two decades ago, Chauve, Dulucq and Guibert showed that the number of rooted
trees on the vertex set in which exactly children of the root are
lower-numbered than the root is . Here I give a
simpler proof of this result.Comment: LaTex2e, 9 pages. Version 2 contains a Note Added with a quick and
elegant proof due to Jiang Zeng. To be published in Discrete Mathematic
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