Classifying descents according to equivalence mod k

Sergey Kitaev, Jeffrey Remmel

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

In an earlier paper the authors refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. In the current paper, we generalize the results of that earlier paper by studying descents according to whether the first or the second element in a descent pair is divisible by k for some k≥2. We provide either an explicit or an inclusion-exclusion type formula for the distribution of the new statistics. Based on our results we obtain combinatorial proofs of a number of remarkable identities. We also provide bijective proofs of some of our results and state a number of open problems.
Original languageEnglish
Article numberR64
Number of pages39
JournalThe Electronic Journal of Combinatorics
Volume13
Publication statusPublished - 2006

Fingerprint

Descent
Equivalence
Statistics
Permutation Statistics
Inclusion-exclusion
Bijective
Divisible
Parity
Open Problems
Generalise

Keywords

  • permutation statistics
  • bijection
  • distribution
  • descents

Cite this

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Classifying descents according to equivalence mod k. / Kitaev, Sergey; Remmel, Jeffrey.

In: The Electronic Journal of Combinatorics, Vol. 13, R64, 2006.

Research output: Contribution to journalArticle

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AB - In an earlier paper the authors refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. In the current paper, we generalize the results of that earlier paper by studying descents according to whether the first or the second element in a descent pair is divisible by k for some k≥2. We provide either an explicit or an inclusion-exclusion type formula for the distribution of the new statistics. Based on our results we obtain combinatorial proofs of a number of remarkable identities. We also provide bijective proofs of some of our results and state a number of open problems.

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