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A082424 Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(n,n) is the Schur function indexed by two parts of size n, s(2n) is the Schur function corresponding to the trivial representation and * represents the inner or Kronecker product. 3

%I #4 Sep 22 2013 15:58:09

%S 1,1,11,41,320,1917,14582,100562,688427,4380888,26324611,148136566,

%T 785175771,3925637781,18586683128,83578440418,358079558873,

%U 1465784048253,5748270468573,21649265291143,78483868584001

%N Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(n,n) is the Schur function indexed by two parts of size n, s(2n) is the Schur function corresponding to the trivial representation and * represents the inner or Kronecker product.

%D I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Oxford Univ. Press, second edition, 1995.

%F a(n) = Sum_{gamma} Chi^{(n, n)}( gamma )^6/z(gamma) the sum is over all partitions gamma of 2n Chi^lambda(gamma) is the value of the symmetric group character z(gamma) is the size of the stablizer of the conjugacy class of symmetric group indexed by the partition gamma

%p compsclr := proc(k) local gamma; add( combinat[Chi]( [k,k], gamma)^6/ZEE(gamma),gamma= combinat[partition](2*k)); end: ZEE := proc (mu) local res, m, i; m := 1; res := convert(mu,`*`); for i from 2 to nops(mu) do if mu[i] <> mu[i-1] then m := 1 else m := m+1 fi; res := res*m; od; res; end:

%Y Cf. A008763 change 6 to 4 in the above program.

%K nonn

%O 0,3

%A _Mike Zabrocki_, Apr 24 2003

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)