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A272090 Number of chiral partitions of n; number of irreducible representations of the symmetric group S_n with nontrivial determinant. 3
0, 1, 2, 3, 5, 4, 8, 12, 20, 8, 16, 24, 40, 32, 64, 88, 152, 16, 32, 48, 80, 64, 128, 192, 320, 128, 256, 384, 640, 512, 1024, 1360, 2384, 32, 64, 96, 160, 128, 256, 384, 640, 256, 512, 768, 1280, 1024, 2048, 2816, 4864, 512, 1024, 1536, 2560, 2048, 4096, 6144, 10240, 4096 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Amritanshu Prasad, Table of n, a(n) for n = 1..9999

Arvind Ayyer, Amritanshu Prasad, Steven Spallone, Representations of symmetric groups with non-trivial determinant, arXiv:1604.08837 [math.RT], 2016.

Arvind Ayyer, Amritanshu Prasad, Steven Spallone, Macdonald trees and determinants of representations for finite Coxeter groups, arXiv:1812.00608 [math.RT], 2018.

Amritanshu Prasad, Sage program

FORMULA

a(n) = A000041(n) - A045923(n).

If n = e + Sum_{i=1..r}2^ki in binary expansion, with e=0 or 1, 0<k1<...<kr, then a(n) = 2^(Sum_{i=2..r}k_i)(2^(k1-1)+Sum_{v=1..k1-1}2^((v+1)(k1-2)-v(v-1)/2)+e2^(k1(k1-1)/2).

EXAMPLE

The sign representation and the two-dimensional representation of S_3 have nontrivial determinant, so a(3)=2.

MATHEMATICA

a[1] = 0;

a[n_] := Module[{bb, e, pos, k, r}, bb = Reverse[IntegerDigits[n, 2]]; e = bb[[1]]; pos = DeleteCases[Flatten[Position[bb, 1]], 1]-1; r = Length[ pos]; Do[k[i] = pos[[i]], {i, r}]; 2^Sum[k[i], {i, 2, r}] (2^(k[1]-1) + Sum[2^((v+1)(k[1]-2)-v(v-1)/2), {v, k[1]-1}] + e 2^(k[1] (k[1]-1)/2))];

Array[a, 60] (* Jean-Fran├žois Alcover, Aug 09 2018 *)

PROG

(PARI) a(n) = {if (n==1, 0, if (n % 2, ns = n-1; eps = 1, ns = n; eps = 0); b = Vecrev(binary(ns/2)); vk = select(x->(x != 0), b, 1); k1 = vk[1]; 2^sum(i=2, #vk, vk[i])*(2^(k1-1) + sum(v=1, k1-1, 2^((v+1)*(k1-2)-binomial(v, 2))) + eps*2^binomial(k1, 2)); ); } \\ Michel Marcus, May 11 2016

CROSSREFS

Cf. A000041, A045923.

Sequence in context: A046708 A185728 A285492 * A120255 A245608 A244154

Adjacent sequences:  A272087 A272088 A272089 * A272091 A272092 A272093

KEYWORD

nonn

AUTHOR

Amritanshu Prasad, May 10 2016

STATUS

approved

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Last modified January 17 18:14 EST 2020. Contains 330987 sequences. (Running on oeis4.)