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

    • Keywords

       

      Set partitions; descents; partition statistic; combinatorial proof.

    • Abstract

       

      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.

    • Author Affiliations

       

      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
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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