Browse issues
no. 1
Descent set distribution for permutations with cycles of only odd or only even lengths
Adin, Ron M.1; Hegedűs, Pál2; Roichman, Yuval1
1Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900 (Israel)
2Department of Algebra and Geometry, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest (Hungary)
Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 161-182

It is known that the number of permutations in the symmetric group $S_{2n}$ with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent sets: the number of permutations in $S_{2n}$ with a prescribed descent set and all cycles of odd lengths is equal to the number of permutations with the complementary descent set and all cycles of even lengths. There is also a variant for $S_{2n+1}$. The proof uses generating functions for character values and applies a new identity on higher Lie characters.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.471
Classification: 05A05, 05E05, 20C30
Keywords: permutation, descent set, higher Lie character, character generating function

Adin, Ron M.  1 ; Hegedűs, Pál  2 ; Roichman, Yuval  1

1 Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900 (Israel)
2 Department of Algebra and Geometry, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest (Hungary)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2026__9_1_161_0,
     author = {Adin, Ron M. and Heged\'{u}s, P\'al and Roichman, Yuval},
     title = {Descent set distribution for permutations with cycles of only odd or only even lengths},
     journal = {Algebraic Combinatorics},
     pages = {161--182},
     year = {2026},
     publisher = {The Combinatorics Consortium},
     volume = {9},
     number = {1},
     doi = {10.5802/alco.471},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.471/}
}
TY  - JOUR
AU  - Adin, Ron M.
AU  - Hegedűs, Pál
AU  - Roichman, Yuval
TI  - Descent set distribution for permutations with cycles of only odd or only even lengths
JO  - Algebraic Combinatorics
PY  - 2026
SP  - 161
EP  - 182
VL  - 9
IS  - 1
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.471/
DO  - 10.5802/alco.471
LA  - en
ID  - ALCO_2026__9_1_161_0
ER  - 
%0 Journal Article
%A Adin, Ron M.
%A Hegedűs, Pál
%A Roichman, Yuval
%T Descent set distribution for permutations with cycles of only odd or only even lengths
%J Algebraic Combinatorics
%D 2026
%P 161-182
%V 9
%N 1
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.471/
%R 10.5802/alco.471
%G en
%F ALCO_2026__9_1_161_0
Adin, Ron M.; Hegedűs, Pál; Roichman, Yuval. Descent set distribution for permutations with cycles of only odd or only even lengths. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 161-182. doi: 10.5802/alco.471

[1] Adin, Ron M.; Hegedűs, Pál; Roichman, Yuval Higher Lie characters and root enumeration in classical Weyl groups, J. Algebra, Volume 664 (2025), pp. 26-72 | DOI | MR | Zbl

[2] Bóna, Miklós A walk through combinatorics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2024, xxi+613 pages | MR | Zbl

[3] Chernoff, W. W. Solutions to x r =α in the alternating group, Ars Combin., Volume 29 (1990), pp. 226-227 Twelfth British Combinatorial Conference (Norwich, 1989) | MR | Zbl

[4] Elizalde, Sergi A bijection for descent sets of permutations with only even and only odd cycles, European J. Combin., Volume 132 (2026), Paper no. 104280, 17 pages | DOI | MR | Zbl

[5] Gessel, Ira M.; Reutenauer, Christophe Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A, Volume 64 (1993) no. 2, pp. 189-215 | DOI | MR | Zbl

[6] Glebsky, Lev; Licón, Melany; Rivera, Luis Manuel On the number of even roots of permutations, Australas. J. Combin., Volume 86 (2023), pp. 308-319 | MR | Zbl

[7] Isaacs, I. Martin Character theory of finite groups, Dover Publications, Inc., New York, 1994, xii+303 pages | MR | Zbl

[8] Leaños, Jesús; Moreno, Rutilo; Rivera-Martínez, Luis Manuel On the number of mth roots of permutations, Australas. J. Combin., Volume 52 (2012), pp. 41-54 | MR | Zbl

[9] OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences (Published electronically at http://oeis.org)

[10] Scharf, Thomas Die Wurzelanzahlfunktion in symmetrischen Gruppen, J. Algebra, Volume 139 (1991) no. 2, pp. 446-457 | DOI | MR | Zbl

[11] Scharf, Thomas Über Wurzelanzahlfunktionen voller monomialer Gruppen (German) [On root number functions of full monomial groups], Bayreuth. Math. Schr., Volume 38 (1991), pp. 99-207 (Dissertation, Universität Bayreuth, Bayreuth, 1991) | MR | Zbl

[12] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR | Zbl

Cited by Sources: