Loehr, Nicholas A. and Niese, Elizabeth - Recursions and divisibility properties for combinatorial Macdonald polynomials

dmtcs:541 - Discrete Mathematics & Theoretical Computer Science, February 15, 2011, Vol. 13 no. 1
Recursions and divisibility properties for combinatorial Macdonald polynomials

Authors: Loehr, Nicholas A. and Niese, Elizabeth

For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.


Source : oai:HAL:hal-00990490v1
Volume: Vol. 13 no. 1
Section: Combinatorics
Published on: February 15, 2011
Submitted on: December 28, 2009
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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