Journal of Combinatorics

Volume 11 (2020)

Number 3

Statistics on ordered partitions of sets

Pages: 557 – 574

DOI: https://dx.doi.org/10.4310/JOC.2020.v11.n3.a8

Author

Einar Steingrímsson (Department of Computer and Information Sciences, University of Strathclyde, Glasgow, Scotland, United Kingdom)

Abstract

We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily. The distribution of these statistics is closely related to the $q$-Stirling numbers of the second kind. Some of the statistics are generalizations of known statistics on set partitions, but others are entirely new. All the new ones are sums of two statistics, inspired by statistics on permutations, where one of the two statistics is based on a certain partial ordering of the blocks of a partition.

Keywords

ordered set partitions, $q$-Stirling numbers, permutation statistics

Full Text (PDF format)

Part of this research was carried out while the author was visiting the Université Louis-Pasteur in Strasbourg in 1996, supported by the EU Network in Algebraic Combinatorics.

Received 20 April 2019

Published 11 May 2020