A321931 - OEIS

A321931

Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in M(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and M is augmented monomial symmetric functions.

1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 2, -3, 1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 2, -1, -2, 1, 0, -6, 3, 8, -6, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 2, -1, -2, 1, 0, 0, 0, 2, -2, -1, 0, 1, 0, 0, -6, 6, 5, -3, -3, 1, 0

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OFFSET

1,12

COMMENTS

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

The augmented monomial symmetric functions are given by M(y) = c(y) * m(y) where c(y) = Product_i (y)_i! where (y)_i is the number of i's in y and m is monomial symmetric functions.

LINKS

Table of n, a(n) for n=1..81.

Wikipedia, Symmetric polynomial

EXAMPLE

Tetrangle begins (zeros not shown):

(1): 1

(2): 1

(11): -1 1

(3): 1

(21): -1 1

(111): 2 -3 1

(4): 1

(22): -1 1

(31): -1 1

(211): 2 -1 -2 1

(1111): -6 3 8 -6 1

(5): 1

(41): -1 1

(32): -1 1

(221): 2 -1 -2 1

(311): 2 -2 -1 1

(2111): -6 6 5 -3 -3 1

(11111): 24 30 20 15 20 10 1

For example, row 14 gives: M(32) = -p(5) + p(32).

CROSSREFS

Row sums are A155972. This is a regrouping of the triangle A321895.

Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

Sequence in context: A175669 A288839 A286583 * A321934 A004579 A081371

Adjacent sequences: A321928 A321929 A321930 * A321932 A321933 A321934

KEYWORD

sign,tabf

AUTHOR

Gus Wiseman, Nov 23 2018

STATUS

approved