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Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.
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%I #15 Dec 20 2018 22:47:16

%S 1,1,1,-1,1,2,3,1,-1,0,1,2,-3,1,6,3,8,6,1,0,3,-4,0,1,-2,-1,0,2,1,2,-1,

%T 0,-2,1,-6,3,8,-6,1,24,30,20,15,20,10,1,-6,0,-5,0,5,5,1,0,-6,4,3,-4,2,

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

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

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

%C We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is the basis of Schur functions and syt(y) is the number of standard Young tableaux of shape y.

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

%e Tetrangle begins (zeros not shown):

%e (1): 1

%e .

%e (2): 1 1

%e (11): -1 1

%e .

%e (3): 2 3 1

%e (21): -1 1

%e (111): 2 -3 1

%e .

%e (4): 6 3 8 6 1

%e (22): 3 -4 1

%e (31): -2 -1 2 1

%e (211): 2 -1 -2 1

%e (1111): -6 3 8 -6 1

%e .

%e (5): 24 30 20 15 20 10 1

%e (41): -6 -5 5 5 1

%e (32): -6 4 3 -4 2 1

%e (221): 6 -4 3 -4 -2 1

%e (311): 4 -5 1

%e (2111): -6 5 5 -5 1

%e (11111): 24 30 20 15 20 10 1

%e For example, row 14 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).

%Y This is a regrouping of the triangle A321900.

%Y Cf. A008480, A056239, A124794, A124795, A153452 (standard Young tableaux), A215366, A296188, A300121, A319191, A319193, A321908, A321912-A321934.

%K sign,tabf

%O 1,6

%A _Gus Wiseman_, Nov 23 2018