A382819 Number of Grassmannian permutations on [n] of order dividing 3.

1, 1, 1, 3, 5, 7, 12, 17, 22, 31, 40, 49, 63, 77, 91, 111, 131, 151, 178, 205, 232, 267, 302, 337, 381, 425, 469, 523, 577, 631, 696, 761, 826, 903, 980, 1057, 1147, 1237, 1327, 1431, 1535, 1639, 1758, 1877, 1996, 2131, 2266, 2401, 2553, 2705, 2857, 3027, 3197, 3367, 3556, 3745, 3934

Offset

0,4

Links

Table of n, a(n) for n=0..56.

Kassie Archer and Aaron Geary, Descents in powers of permutations, arXiv:2406.09369 [math.CO], 2024.

Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Formula

a(n) = binomial(floor(n/3)+3,3) + binomial(floor((n-1)/3)+3,3) + binomial(floor((n-2)/3)+3,3) - n.

a(n) = 2*a(n-1) - a(n-2) + n/3 + 1 for n mod 3 = 0

a(n) = 2*a(n-1) - a(n-2) for n mod 3 <> 0.

a(n) ~ n^3/54. - Stefano Spezia, Apr 06 2025

G.f.: -(x^7-2*x^4+x-1)/((x^2+x+1)^2*(x-1)^4). - Alois P. Heinz, Apr 06 2025

Example

For n = 4 there are 5 Grassmannian permutations whose cubes are the identity permutation: 1234, 3124, 1423, 2314, 1342, so a(4) = 5.

Crossrefs

Cf. A000325 (Grassmannian permutations), A001470 (permutations of order dividing 3).

Sequence in context: A241515 A070334 A137700 * A325267 A241544 A208716

Adjacent sequences: A382816 A382817 A382818 * A382820 A382821 A382822

Keyword

nonn,easy

Author

Aaron Geary, Apr 05 2025

Status

approved