Taiwanese Journal of Mathematics

Enumerations of Permutations by Circular Descent Sets

Hungyung Chang, Jun Ma, and Jean Yeh

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Abstract

The circular descent of a permutation $\sigma$ is a set $\{ \sigma(i) \mid \sigma(i) \gt \sigma(i+1) \}$. In this paper, we focus on the enumerations of permutations by the circular descent set. Let $\operatorname{cdes}_n(S)$ be the number of permutations of length $n$ which have the circular descent set $S$. We derive the explicit formula for $\operatorname{cdes}_n(S)$. We describe a class of generating binary trees $T_k$ with weights. We find that the number of permutations in the set $\operatorname{CDES}_n(S)$ corresponds to the weights of $T_k$. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 13 pages.

Dates
First available in Project Euclid: 28 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1548666174

Digital Object Identifier
doi:10.11650/tjm/190105

Subjects
Primary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

Keywords
circular descent generating tree permutation permutation tableaux

Citation

Chang, Hungyung; Ma, Jun; Yeh, Jean. Enumerations of Permutations by Circular Descent Sets. Taiwanese J. Math., advance publication, 28 January 2019. doi:10.11650/tjm/190105. https://projecteuclid.org/euclid.twjm/1548666174


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References

  • F. R. K. Chung, R. L. Graham, V. E. Hoggat, Jr. and M. Kleiman, The number of Baxter permutations, J. Combin. Theory Ser. A 24 (1978), no. 3, 382–394.
  • S. Corteel, Crossings and alignments of permutations, Adv. in Appl. Math. 38 (2007), no. 2, 149–163.
  • S. Corteel and P. Nadeau, Bijections for permutaion tableaux, European J. Combin. 30 (2009), no. 1, 295–310.
  • S. Corteel and L. K. Williams, Permutation tableaux and the asymmetric exclusion process, Adv. in Appl. Math. 39 (2006), no. 3, 293–310.
  • ––––, A Markov chain on permutations which projects to the PASEP, Int. Math. Res. Not. IMRN 2007 (2007), no. 17, Art. ID rnm055, 27 pp.
  • M. Domaratzki, Combinatorial Interpretations of a generalization of the Genocchi numbers, J. Integer Seq. 7 (2004), no. 3, Article 04.3.6, 11 pp.
  • D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Math. 1 (1972), no. 4, 321–327.
  • ––––, Interprétations combinatoires des nombres de Genocchi, Duke Math. J. 41 (1974), 305–318.
  • D. Dumont and D. Foata, Une propriété de symétrie des nombres de Genocchi, Bull. Soc. Math. France 104 (1976), no. 4, 433–451.
  • D. Dumont and A. Randrianarivony, Dérangements et nombres de Genocchi, Discrete Math. 132 (1994), no. 1-3, 37–49.
  • D. Dumont and G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Ann. Discrete. Math. 6 (1980), 77–87.
  • J. M. Gandhi, A conjectured representation of Genocchi numbers, Amer. Math. Monthly 77 (1970), no. 5, 505–506.
  • A. Postnikov, Total positivity, Grassmannians, and networks..
  • J. Riordan and P. R. Stein, Proof of a conjecture on Genocchi numbers, Discrete Math. 5 (1973), 381–388.
  • E. Steingrimsson and L. K. Willams, Permutation tableaux and permutation patterns, J. Combin. Theory Ser. A 114 (2007), no. 2, 211–234.
  • L. K. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005), no. 2, 319–342.