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A321935 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.

%I A321935 #15 Dec 20 2018 22:47:16
%S A321935 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 A321935 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 A321935 1,0,6,-4,3,-4,-2,1,4,0,0,-5,0,0,1,-6,0,5,0,5
%N A321935 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 A321935 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321935 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 A321935 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e A321935 Tetrangle begins (zeros not shown):
%e A321935   (1):  1
%e A321935 .
%e A321935   (2):   1  1
%e A321935   (11): -1  1
%e A321935 .
%e A321935   (3):    2  3  1
%e A321935   (21):  -1     1
%e A321935   (111):  2 -3  1
%e A321935 .
%e A321935   (4):     6  3  8  6  1
%e A321935   (22):       3 -4     1
%e A321935   (31):   -2 -1     2  1
%e A321935   (211):   2 -1    -2  1
%e A321935   (1111): -6  3  8 -6  1
%e A321935 .
%e A321935   (5):     24 30 20 15 20 10  1
%e A321935   (41):    -6    -5     5  5  1
%e A321935   (32):       -6  4  3 -4  2  1
%e A321935   (221):       6 -4  3 -4 -2  1
%e A321935   (311):    4       -5        1
%e A321935   (2111):  -6     5     5 -5  1
%e A321935   (11111): 24 30 20 15 20 10  1
%e A321935 For example, row 14 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).
%Y A321935 This is a regrouping of the triangle A321900.
%Y A321935 Cf. A008480, A056239, A124794, A124795, A153452 (standard Young tableaux), A215366, A296188, A300121, A319191, A319193, A321908, A321912-A321934.
%K A321935 sign,tabf
%O A321935 1,6
%A A321935 _Gus Wiseman_, Nov 23 2018