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A097567 T(n,k)= count of partitions p such that Abs( Odd(p)-Odd(p') ) = k, where p' is the transpose of p and Odd(p) counts the odd elements in p. Related to Stanley's 'f'.

%I A097567 #11 Nov 29 2018 03:02:58
%S A097567 1,1,0,0,0,2,1,0,2,0,3,0,0,0,2,3,0,2,0,2,0,1,0,8,0,0,0,2,3,0,8,0,2,0,
%T A097567 2,0,10,0,2,0,8,0,0,0,2,10,0,8,0,8,0,2,0,2,0,4,0,26,0,2,0,8,0,0,0,2,
%U A097567 10,0,26,0,8,0,8,0,2,0,2,0,27,0,10,0,28,0,2,0,8,0,0,0,2,27,0,26,0,28,0,8,0,8
%N A097567 T(n,k)= count of partitions p such that Abs( Odd(p)-Odd(p') ) = k, where p' is the transpose of p and Odd(p) counts the odd elements in p. Related to Stanley's 'f'.
%C A097567 Table starts {1}, {1,0}, {0,0,2}, {1,0,2,0}, {3,0,0,0,2}, .. where the odd columns are 0. Row sums are A000041 by definition.
%H A097567 George E. Andrews, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i2r1">On a Partition Function of Richard Stanley</a>, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R1.
%H A097567 Andrew V. Sills, <a href="http://dx.doi.org/10.1155/S0161171204401380">A Combinatorial proof of a partition identity of Andrews and Stanley</a>, International Journal of Mathematics and Mathematical Sciences, Volume 2004, Issue 47, Pages 2495-2501.
%t A097567 Table[par=Partitions[n];Table[Count[par, q_/;Abs[Count[q, _?OddQ]-Count[TransposePartition[q], _?OddQ]]===k], {k, 0, n}], {n, 0, 16}]
%Y A097567 Cf. A097566.
%K A097567 easy,nonn,tabl
%O A097567 0,6
%A A097567 _Wouter Meeussen_, Aug 28 2004