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Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in p(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.
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%I #4 Nov 22 2018 18:18:13

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

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

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

%N Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in p(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

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

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

%e Tetrangle begins (zeroes not shown):

%e (1): 1

%e .

%e (2): -2 1

%e (11): 1

%e .

%e (3): 3 -3 1

%e (21): -2 1

%e (111): 1

%e .

%e (4): -4 2 4 -4 1

%e (22): 4 -4 1

%e (31): 3 -3 1

%e (211): -2 1

%e (1111): 1

%e .

%e (5): 5 -5 -5 5 5 -5 1

%e (41): -4 2 4 -4 1

%e (32): -6 6 3 -5 1

%e (221): 4 -4 1

%e (311): 3 -3 1

%e (2111): -2 1

%e (11111): 1

%e For example, row 14 gives: p(32) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).

%Y This is a regrouping of the triangle A321752.

%Y Cf. A005651, A008480, A056239, A124794, A124795, A135278, A215366, A318284, A319191, A319193, A319225, A319226, A321912-A321935.

%K sign,tabf

%O 1,2

%A _Gus Wiseman_, Nov 22 2018