[seqfan] mathoverflow as a source of inspiration

Wouter Meeussen wouter.meeussen at telenet.be
Sun Oct 21 00:24:39 CEST 2012


hi all,

on http://mathoverflow.net/questions/108890/tensor-powers-of-the-standard-representation
the question was asked :
“Consider V(n-1,1) , the n-1 dimensional irreducible representation of (the Symmetric group) Sn, i.e. the "standard" or "defining" representation. Is there a nice formula for how its k-th tensor power decomposes into irreps?”

The replies are quite involuted. But specialising to a simpler case seems to tie up a bunch of known sequences:

“Consider V(n-1,1) , the n-1 dimensional irreducible representation of Sn, i.e. the "standard" or "defining" representation. Is there a nice formula for the coefficient of V(n-1,1) in its k-th tensor power?”

(* note that ‘tensor power’ can be replaced with just ‘power’ when characters are seen as just a list *)

I found that the coefficient of V(n-1,1) in V(n-1,1)^k is :
n=2:
%S A000035 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1
          %N A000035 Period 2: (0, 1) repeated; a(n) = n mod 2.
n=3:
%S A001045 0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691
          %N A001045 Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1.
n=4:
%S A006342 1,1,4,10,31,91,274,820,2461,7381,22144,66430,199291,597871,1793614
          %N A006342 Coloring a circuit with 4 colors.
n=5:
%S A214142 1,1,4,11,40,147,568,2227,8824,35123,140152,559923,2238328,8950579
          %N A214142 Number of 0..4 colorings of a 1X(n+1) array circular in the n+1 direction with new values 0..4 introduced in row major order
n=6:
%S A214167 1,1,4,11,41,161,694,3151,14851,71621,350384,1729091,8577661,42686281
          %N A214167 Number of 0..5 colorings of a 1X(n+1) array circular in the n+1 direction with new values 0..5 introduced in row major order
n=7:
%S A214188 1,1,4,11,41,162,714,3397,17251,92048,509444,2893683,16734381    
          %N A214188 Number of 0..6 colorings of a 1X(n+1) array circular in the n+1 direction with new values 0..6 introduced in row major order
n=8:
%S A214239 1,1,4,11,41,162,715,3424,17686,97493,567986,3462537,21880951    
          %N A214239 Number of 0..7 colorings of a 1X(n+1) array circular in the n+1 direction with new values 0..7 introduced in row major order

and for n large, seems to converge to
%S A000296 1,0,1,1,4,11,41,162,715,3425,17722,98253,580317,3633280,24011157
%N A000296 Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.
%F A000296 E.g.f.: exp(exp(x) - 1 - x).

ps. check the example for n=4 in http://www.mat.univie.ac.at/~slc/wpapers/s54goupchau.html

I hope someone more knowledgeable in group theory will come up with a validation the above observations, and an easy–to-understand comment to be added to these sequences.

Wouter.

------------------------- Mathematica 4.1 implementation --------------------------------
uses definitions in  http://users.telenet.be/Wouter.Meeussen/ToolBox.nb

Table[(chars[{n - 1, 1}]^k cycleclasses[n]/n!). chars[{n - 1, 1}], {n, 2, 
      12}, {k, 12}] // MatrixForm
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