Difference operators for partitions and some applications
Abstract
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 $k\in \mathbb{N}$. 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 $D^-$ defined on functions of partitions. Even though the calculations for higher orders of $D$ are extremely complex, we prove that several well-known 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 $K_r$ arise directly from the computation for a single partition $\lambda$, without the summation ranging over all partitions of size~$n$.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2015
- DOI:
- 10.48550/arXiv.1508.00772
- arXiv:
- arXiv:1508.00772
- Bibcode:
- 2015arXiv150800772H
- Keywords:
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- Mathematics - Combinatorics;
- 05A15;
- 05A17;
- 05A19;
- 05E05;
- 05E10;
- 11P81
- E-Print:
- 27 pages