A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

1, 2, 8, 26, 94, 326, 1196, 4358, 16248, 60854, 230184, 874878, 3343614, 12825418, 49368388, 190554410, 737328366, 2858974502, 11106267880, 43215101102, 168398785002, 657070401106, 2566847255572, 10038191414610, 39295007540748

Offset: 0

Author

Richard Stanley, Feb 15 2002

Keywords

• easy
• nonn

Comments

Original name: "Squared length of sum of s_lambda^2, where s_lambda is a Schur function and lambda ranges over all partitions of n."

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4. - Seiichi Manyama, Apr 22 2018

The symbol "|-" means "is a partition of", cf. MathWorld link and the Geloun & Ramgoolam paper. The Littlewood-Richardson coefficients allow a product of two Schur functions to be expressed as a linear combination of Schur functions of the corresponding degree. (The Schur functions symmetric in all n variables correspond to Schur polynomials of partitions extended with 0's to length n.) - M. F. Hasler, Jan 19 2020

See A070933 for similar sums of squares of Littlewood-Richardson coefficients. - M. F. Hasler, Jan 20 2020

Examples

			For n=3 the s_lambda^2 summed over all partitions of n and decomposed into a sum of Schur functions yields
    s(6) + 2 s(3,3) + 2 s(4,2) + s(5,1) + 2 s(2,2,2) + 2 s(3,2,1) + s(4,1,1)
    + 2 s(2,2,1,1) + s(3,1,1,1) + s(2,1,1,1,1) + s(1,1,1,1,1,1),
  and the sum of the squares of the coefficients {1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1} gives a(3) = 26.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
• Eric Weisstein's World of Mathematics, Partition.
• Wikipedia, Littlewood-Richardson rule, as of Dec 18 2018.
• Wikipedia, Schur polynomial, as of Jan 13 2020.

Crossrefs

Cf. A001868.

List of partitions: A036037, A080577, A181317, A330370.

Cf. A070933 (Sum_{lambda,mu,nu} (c^{lambda}_{mu,nu})^2, |mu| = |nu| = n).

Cf. A003040 (maximum number of standard tableaux of the Ferrers diagrams of the partitions of n).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      binomial(n+n, n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 24 2019

Mathematica

Table[Tr[(Apply[List,
  Sum[Tr[s @@@ LRRule[\[Lambda], \[Lambda]]],
   {\[Lambda], Partitions[n]}]] /. s[] -> 1)^2], {n, 1, 10}];
(* with 'LRRule' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb - Wouter Meeussen, Jan 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Binomial[n+n, n],
     Sum[b[j, 1]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 02 2022, after Alois P. Heinz *)

PARI

A067855_upto(N)=Vec(1/sqrt(prod(i=1,N-1,1-4*'x^i+O('x^N)))) \\ M. F. Hasler, Jan 23 2020

Formula

G.f.: 1/sqrt(Product_{i >= 1} (1 - 4*x^i)).

Euler transform of A001868(n)/2. a(n) = Sum_{pi} Product_{m=1..n} binomial(2*p(m), p(m)), where pi runs through all nonnegative solutions of p(1) + 2*p(2) + ... + n*p(n)=n. - Vladeta Jovovic, Mar 25 2006

a(n) ~ 2^(2*n) / sqrt(c*Pi*n), where c = QPochhammer[1/4] = 0.688537537120339... - Vaclav Kotesovec, Apr 22 2018

By definition, a(n) = Sum_{mu |- 2n} c_mu^2 where Sum_{lambda |- n} s_lambda^2 = Sum_{mu |- 2n} c_mu s_mu, where s_lambda are the Schur polynomials (symmetric in 2n variables) and the sums run over all partitions of n resp. 2n. - M. F. Hasler, Jan 19 2020

Extensions

More terms from Vladeta Jovovic, Mar 25 2006

Name edited by M. F. Hasler following observations by Wouter Meeussen, Jan 17 2020