A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten 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

1,12

Comments

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

The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

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: F(32) = -p(5) - p(32).

Crossrefs

Row sums are A178803. Up to sign, same as A321931. This is a regrouping of the triangle A321899.

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

Sequence in context: A288839 A286583 A321931 * A004579 A081371 A321935

Adjacent sequences: A321931 A321932 A321933 * A321935 A321936 A321937

Keyword

sign,tabf

Author

Gus Wiseman, Nov 23 2018

Status

approved