Difference Operators for Partitions and Some Applications

被引:0
|
作者
Guo-Niu Han
Huan Xiong
机构
[1] I.R.M.A.,
[2] UMR 7501,undefined
[3] Université de Strasbourg et CNRS,undefined
来源
Annals of Combinatorics | 2018年 / 22卷
关键词
partition; hook length; content; standard Young tableau; difference operator; 05A15; 05A17; 05A19; 05E05; 05E10; 11P81;
D O I
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中图分类号
学科分类号
摘要
Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the 2k-th power sum of hook lengths of partitions with size n is always a polynomial of n for any k∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k \in \mathbb{N}}$$\end{document}. This conjecture was generalized and proved by Stanley (Ramanujan J. 23(1–3), 91–105 (2010)). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators D and D-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D^{-}}$$\end{document} defined on functions of partitions. Even though the calculations for higher orders of D are extremely complex, we prove that several wellknown families of functions of partitions are annihilated by a power of the difference operator D. As an application, our results lead to several generalizations of classic results on partitions, including the marked hook formula, Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula. We insist that the Okada constants Kr arise directly from the computation for a single partition λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document}, without the summation ranging over all partitions of size n.
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页码:317 / 346
页数:29
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