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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.
23
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, 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, 1, 0, 6, -4, 3, -4, -2, 1, 4, 0, 0, -5, 0, 0, 1, -6, 0, 5, 0, 5
OFFSET
1,6
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
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.
EXAMPLE
Tetrangle begins (zeros not shown):
(1): 1
.
(2): 1 1
(11): -1 1
.
(3): 2 3 1
(21): -1 1
(111): 2 -3 1
.
(4): 6 3 8 6 1
(22): 3 -4 1
(31): -2 -1 2 1
(211): 2 -1 -2 1
(1111): -6 3 8 -6 1
.
(5): 24 30 20 15 20 10 1
(41): -6 -5 5 5 1
(32): -6 4 3 -4 2 1
(221): 6 -4 3 -4 -2 1
(311): 4 -5 1
(2111): -6 5 5 -5 1
(11111): 24 30 20 15 20 10 1
For example, row 14 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).
CROSSREFS
This is a regrouping of the triangle A321900.
Sequence in context: A321934 A004579 A081371 * A309936 A321900 A352195
KEYWORD
sign,tabf
AUTHOR
Gus Wiseman, Nov 23 2018
STATUS
approved