A272090 Number of chiral partitions of n; number of irreducible representations of the symmetric group S_n with nontrivial determinant.
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
Offset: 1
Keywords
Examples
The sign representation and the two-dimensional representation of S_3 have nontrivial determinant, so a(3)=2.
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
Programs
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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-François Alcover, Aug 09 2018 *) -
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