• Chunwei Song

Articles written in Proceedings – Mathematical Sciences

• Enumerating Set Partitions According to the Number of Descents of Size 𝑑 or more

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.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

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Posted on July 25, 2019