[seqfan] A067855 : looking for an explanation of the title

Wouter Meeussen wouter.meeussen at telenet.be
Fri Jan 17 19:41:07 CET 2020

A067855 : Squared length of sum of s_lambda^2, where s_lambda is a Schur function and lambda ranges over all partitions of n.
for n=0 to 7 : a(n) = 1, 2, 8, 26, 94, 326, 1196, 4358

I’m trying to understand this title by working out an example on n=3 giving a(3) = 26.

The only way I see to get the sequence 1, 2, 8, 26, 94 is by counting the number of terms in the Littlewood-Richardson expansion of
s(lambda) s(mu)  with lambda and mu partitions of 3 AND lambda >= mu.
This amounts to the lower diagonal matrix of the number of terms (including multiplicities) in s(lambda)*s(mu) .

BUT, using this method, for n=5, 6, 7 we get 
306, 1048, 3370 and not the claimed 326, 1196, 4358.

Since the original submission was by R. P. Stanley in 2002 and included the G.f. it must be correct.
I checked his Enumerative Combinatorics Vol. 2, chapter 7 and its exercises, but found no hints to the above.

Can anyone cook up an example to what is meant by “squared length” of a sum of “s^2”?


... all lost in darkness ...

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