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

Offset: 1

Author

Gus Wiseman, Nov 23 2018

Keywords

• sign
• tabf

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.

Examples

			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).
		

Links

• Wikipedia, Symmetric polynomial

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.