[seqfan] a strange but nice hit by superseeker

[Wouter Meeussen]

dear Sequence fans,

I tested this partitions related sequence with superseeker:

a(n) = 1, 0, 1, 1, 3, 5, 9, 14, 22, 32, 46, 65, 89, 121, 162, ...  offset 1

where a(n) equals the sum of the n-th 'quotient sign' of lambda(n) over 
all partitions lambda of n.
This 'quotient sign' is defined in 
http://web.math.ku.dk/~olsson/manus/comb_rep_all.pdf 
<http://web.math.ku.dk/%7Eolsson/manus/comb_rep_all.pdf> (pg 22, example 
following remark 3.11).
It is defined for a partition lambda of n and an integer k <= n. I have 
sofar not been able to find a better reference for it. Pointers are 
appreciated.
For the Mathematica affectionados I attach a function for it at the end.

Example:
the quotient signs of {3,2,1,1} for k=1 to 7 are
{1, -1, -1, 1, 1, -1, 1}
and the 7-th quotient sign of the partitions of 7 are
{1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, 1} summing to 9.

Superseeker found that a(n)+n-2 apparently equals A325269 ( Number of 
integer partitions of n with 2 distinct parts or at least 3 parts. 
AUTHOR Gus Wiseman <https://oeis.org/wiki/User:Gus_Wiseman>, Apr 18 2019).
Tested up to n=46.

other strange observation :

q(lambda,n), the n-th quotient sign of lambda(n), seems to also satisfy
del(lambda,n) q(lambda,n) = s(lambda, x_1,x_2, .. ,x_n) for x_k = exp(2 
pi i k/n)
with del(lambda,n) = 1 iff the n-core of lambda(n) is empty, else 0
and s(lambda) the Schur polynomial in n variables. Tested up tp n=20.

weird!

Wouter.

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quotientsign[\[Lambda]_?PartitionQ, k_Integer] /; (k <= Tr[\[Lambda]]) :=
Block[{X, abac, ta, w1, w2},
   X = Reverse[First /@ hooklength[\[Lambda]]];
   abac = Transpose[    Partition[Range[0, First[\[Lambda]] + 
Length[\[Lambda]] + k - 1], k]];
   ta = Sort[Flatten[Position[abac, #] & /@ X, 1]];
   w1 = PadRight at SplitBy[ta, First];
   w2 = Extract[abac, DeleteCases[Flatten[Transpose[w1], 1], {0, 0}]];
   Signature[Flatten[Position[X, #] & /@ w2]]];

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