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Irregular triangle where T(n,k) is the sum of coefficients in the expansion of p(y) in terms of Schur functions, where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).
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%I #12 Sep 16 2018 21:35:19

%S 1,0,2,1,0,4,0,2,1,0,10,1,0,0,2,2,0,26,0,0,1,4,0,0,0,4,4,0,76,1,0,0,0,

%T 0,2,2,4,0,0,0,8,10,0,232,0,1,0,4,0,1,0,0,0,0,12,0,4,2,8,0,0,0,20,26,

%U 0,764,1,0,0,0,2,0,0,4,2,0,0,1,10,0,0,0,0

%N Irregular triangle where T(n,k) is the sum of coefficients in the expansion of p(y) in terms of Schur functions, where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

%C Is this sequence nonnegative? If so, is there a combinatorial interpretation?

%e Triangle begins:

%e 1

%e 0 2

%e 1 0 4

%e 0 2 1 0 10

%e 1 0 0 2 2 0 26

%e 0 0 1 4 0 0 0 4 4 0 76

%e 1 0 0 0 0 2 2 4 0 0 0 8 10 0 232

%e A215366(6,4) = 25 corresponds to the partition (33). Since p(33) = s(6) + 2 s(33) - s(51) + 2 s(222) - 2 s(321) + s(411) + s(3111) - s(21111) + s(111111) has sum of coefficients 1 + 2 - 1 + 2 - 2 + 1 + 1 - 1 + 1 = 4, we conclude T(6,4) = 4.

%Y Last column is A000085. Row sums are A082733.

%Y Cf. A056239, A093641, A153452, A153734, A215366, A296188, A296561, A299699, A305940, A317554.

%K nonn,tabf

%O 1,3

%A _Gus Wiseman_, Sep 14 2018