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

[Wouter Meeussen]

Found it:

for n=3, the s_lambda^2 summed over all partitions of n are
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]
with coefficients
{1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1}
and sum of their squares equals 26.

Permission to add this example to A067855 ?

Wouter.

ps. in my Mma dialect, the program would be
Table[Tr[(Apply[List,Sum[Tr[s @@@ LRRule[\[Lambda], \[Lambda]]], {\[Lambda], 
Partitions[n]}]] /. s[__] -> 1)^2], {n, 1, 7}]
but that uses 'LRRule' from my private 
http://users.telenet.be/Wouter.Meeussen/ToolBox.nb


-----Original Message----- 
From: Wouter Meeussen
Sent: Friday, January 17, 2020 7:41 PM
To: seqfan
Subject: [seqfan] A067855 : looking for an explanation of the title

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”?

Wouter.

... all lost in darkness ...

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