科研之友
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摘要
Oscillating tableaux are certain walks in Young's lattice of partitions; they generalize standard Young tableaux. The shape of an oscillating tableau is the last partition it visits and the length of an oscillating tableau is the number of steps it takes. We define a new statistic for oscillating tableaux that we call weight: the weight of an oscillating tableau is the sum of the sizes of all the partitions that it visits. We show that the average weight of all oscillating tableaux of shape A and length vertical bar lambda vertical bar +2n (where vertical bar lambda vertical bar denotes the size of A and n is an element of N) has a surprisingly simple formula: it is a quadratic polynomial in vertical bar lambda vertical bar and n. Our proof via the theory of differential posets is largely computational. We suggest how the homomesy paradigm of Propp and Roby may lead to a more conceptual proof of this result and reveal a hidden symmetry in the set of perfect matchings.
- 出版日期2015-6-15
- 单位MIT
