A321888 -id:A321888 - cp's OEIS frontend

A321889 Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

1, 1, -1, 3, 1, -2, -1, 10, 3, 2, 1, -7, -1, -2, -2, 47, 1, 6

Offset: 1

Gus Wiseman, Nov 20 2018

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

			The sum of coefficients of p(211) = -f(4) - 2f(22) - 2f(31) - 2f(211) is a(12) = -7.

Wikipedia, Symmetric polynomial

Row sums of A321888. An unsigned version is A321751.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A319193, A321742-A321765.

A321928 Tetrangle where T(n,H(u),H(v)) is the coefficient of f(v) in p(u), where u and v are integer partitions of n, H is Heinz number, f is forgotten symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 2, 1, 0, 0, -1, -1, 0, 1, 3, 6, -1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, -1, -2, -2, -2, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 1, 2, 1, 0, 2, 0, 0, -1, -3, -4, -6, -6, -6

Offset: 1

Gus Wiseman, Nov 22 2018

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

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -1
  (11):  1  2
.
  (3):    1
  (21):  -1 -1
  (111):  1  3  6
.
  (4):    -1
  (22):    1  2
  (31):    1     1
  (211):  -1 -2 -2 -2
  (1111):  1  6  4 12 24
.
  (5):      1
  (41):    -1 -1
  (32):    -1    -1
  (221):    1  1  2  2
  (311):    1  2  1     2
  (2111):  -1 -3 -4 -6 -6 -6
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: p(32) = -f(5) - f(32).

Wikipedia, Symmetric polynomial

An unsigned version is A321917. This is a regrouping of the triangle A321888.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

cp's OEIS Frontend

A321889 Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

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