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Order ideals in weak subposets of Young’s lattice and associated unimodality conjectures
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|Title: ||Order ideals in weak subposets of Young’s lattice and associated unimodality conjectures|
|Authors: ||Morse, Jennifer|
|Issue Date: ||7-May-2004|
|Citation: ||Retrieved October 30, 2007 from http://xxx.lanl.gov/find/grp_math/1/au:+Morse_J/0/1/0/all/0/1|
|Abstract: ||The k-Young lattice Y k is a weak subposet of the Young lattice containing partitions whose
first part is bounded by an integer k > 0. The Y k poset was introduced in connection with generalized
Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine
symmetric group ˜ Sk+1 by a maximal parabolic subgroup. We prove a number of properties for Y k
including that the covering relation is preserved when elements are translated by rectangular partitions
with hook-length k. We highlight the order ideal generated by an m× n rectangular shape. This order
ideal, Lk(m, n), reduces to L(m, n) for large k, and we prove it is isomorphic to the induced subposet
of L(m, n) whose vertex set is restricted to elements with no more than k−m+1 parts smaller than m.
We provide explicit formulas for the number of elements and the rank-generating function of Lk(m, n).
We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications
connect to recent work on sieved q-binomial coefficients.|
|Appears in Collections:||Faculty Research and Publications (Mathematics)|
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