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Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and h is homogeneous symmetric functions.
3

%I #5 Nov 20 2018 16:31:03

%S 1,1,2,-1,0,1,3,-3,1,0,2,-1,4,-2,-4,4,-1,0,0,1,0,4,0,-4,1,0,0,3,-3,1,

%T 5,-5,-5,5,5,-5,1,0,0,0,2,-1,6,-6,-6,-3,2,6,12,-9,-6,6,-1,0,4,0,-2,-4,

%U 4,-1,0,0,6,-6,-3,5,-1,0,0,0,0,1,7,-7,-7,-7,14

%N Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and h is homogeneous symmetric functions.

%C Row n has length A000041(A056239(n)).

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%C Up to sign, same as A321752.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>

%e Triangle begins:

%e 1

%e 1

%e 2 -1

%e 0 1

%e 3 -3 1

%e 0 2 -1

%e 4 -2 -4 4 -1

%e 0 0 1

%e 0 4 0 -4 1

%e 0 0 3 -3 1

%e 5 -5 -5 5 5 -5 1

%e 0 0 0 2 -1

%e 6 -6 -6 -3 2 6 12 -9 -6 6 -1

%e 0 4 0 -2 -4 4 -1

%e 0 0 6 -6 -3 5 -1

%e 0 0 0 0 1

%e 7 -7 -7 -7 14 7 7 7 -7 -7 -21 14 7 -7 1

%e 0 0 0 4 0 -4 1

%e For example, row 15 gives: p(32) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

%Y Row sums are all equal to 1.

%Y Cf. A005651, A008480, A056239, A124794, A124795, A135278, A319193, A319225, A319226, A321742-A321765, A321854.

%K sign,tabf

%O 1,3

%A _Gus Wiseman_, Nov 20 2018