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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'.
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%I #11 Nov 29 2018 03:02:58

%S 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 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 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 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 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 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 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 Table[par=Partitions[n];Table[Count[par, q_/;Abs[Count[q, _?OddQ]-Count[TransposePartition[q], _?OddQ]]===k], {k, 0, n}], {n, 0, 16}]

%Y Cf. A097566.

%K easy,nonn,tabl

%O 0,6

%A _Wouter Meeussen_, Aug 28 2004