Taiwanese Journal of Mathematics

Enumerations of Permutations by Circular Descent Sets

Hungyung Chang, Jun Ma, and Jean Yeh

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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

Taiwanese J. Math., Volume 23, Number 6 (2019), 1303-1315.

Received: 19 November 2018
Revised: 10 January 2019
Accepted: 15 January 2019
First available in Project Euclid: 28 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

circular descent generating tree permutation permutation tableaux


Chang, Hungyung; Ma, Jun; Yeh, Jean. Enumerations of Permutations by Circular Descent Sets. Taiwanese J. Math. 23 (2019), no. 6, 1303--1315. doi:10.11650/tjm/190105. https://projecteuclid.org/euclid.twjm/1548666174

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  • 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.