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https://www.ias.ac.in/article/fulltext/pmsc/122/04/0507-0517

Set partitions; descents; partition statistic; combinatorial proof.

Let $P(n,k)$ denote the set of partitions of $\{1,2,\ldots,n\}$ having exactly 𝑘 blocks. In this paper, we find the generating function which counts the members of $P(n,k)$ according to the number of descents of size 𝑑 or more, where $d\geq 1$ is fixed. An explicit expression in terms of Stirling numbers of the second kind may be given for the total number of such descents in all the members of $P(n,k)$. We also compute the generating function for the statistics recording the number of ascents of size 𝑑 or more and show that it has the same distribution on $P(n,k)$ as the prior statistics for descents when $d\geq 2$, by both algebraic and combinatorial arguments.

Toufik Mansour¹ Mark Shattuck¹ Chunwei Song²

1. Department of Mathematics, University of Haifa, 31905 Haifa, Israel
2. School of Mathematical Sciences, LMAM, Peking University, Beijing 100871, People’s Republic of China

Published

November 2012

Manuscript received

1 August 2011