A360808 Number of double cosets of the Sylow 2-subgroup of the symmetric group S_n.

1, 1, 2, 2, 4, 8, 35, 16, 51, 145, 1112, 1145, 10929, 41400, 542785, 40384, 583169, 2781808, 48558706, 65461347, 1277941540, 7370563251, 159694747220, 63387056365, 1500631724572, 10152855622657

Offset: 1

Author

Richard Stanley, Feb 21 2023

Keywords

• nonn
• more

Comments

Let S denote a Sylow 2-subgroup of the symmetric group S_n. Then a(n) is the number of double cosets SwS. [Corrected by Benjamin Sambale, Mar 08 2025]

Links

• Persi Diaconis, Eugenio Giannelli, Robert M. Guralnick, Stacey Law, Gabriel Navarro, Benjamin Sambale, and Hunter Spink, On the number and sizes of double cosets of Sylow subgroups of the symmetric group, arXiv:2504.01149 [math.GR], 2025. See pp. 5, 15.

Formula

Define a symmetric function T_k recursively by T_0 = p_1 (power sum), and T_k is the plethysm h_2[T_{k-1}] for k>0. If n has the binary expansion 2^{a_0} + 2^{a_1} + ..., then set $U_n = T_{a_0}T_{a_1}... Then a_n = (usual scalar product on symmetric functions).

Extensions

a(2) and a(5) corrected by Benjamin Sambale, Mar 08 2025