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A note on statistical averages for oscillating tableaux

Author(s)
Hopkins, Samuel Francis; Zhang, Ingrid
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Abstract
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 λ and length |λ| + 2n (where |λ| denotes the size of λ and n ∈ N) has a surprisingly simple formula: it is a quadratic polynomial in |λ| 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.
Date issued
2015-06
URI
http://hdl.handle.net/1721.1/98411
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Electronic Journal of Combinatorics
Publisher
European Mathematical Information Service (EMIS)
Citation
Hopkins, Sam, and Ingrid Zhang. "A note on statistical averages for oscillating tableaux." The Electronic Journal of Combinatorics 22(2) (2015), #P2.48.
Version: Final published version
ISSN
1077-8926

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