A321916 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in h(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

1, -1, 1, 0, 1, 1, -2, 1, 0, -1, 1, 0, 0, 1, -1, 1, 2, -3, 1, 0, 1, 0, -2, 1, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 1, -2, -2, 3, 3, -4, 1, 0, -1, 0, 1, 2, -3, 1, 0, 0, -1, 2, 1, -3, 1, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, -1, 1

Offset: 1

Author

Gus Wiseman, Nov 22 2018

Keywords

• sign
• tabf

Comments

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

Also the coefficient of h(v) in e(u).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -1  1
  (11):     1
.
  (3):    1 -2  1
  (21):     -1  1
  (111):        1
.
  (4):    -1  1  2 -3  1
  (22):       1    -2  1
  (31):          1 -2  1
  (211):           -1  1
  (1111):              1
.
  (5):      1 -2 -2  3  3 -4  1
  (41):       -1     1  2 -3  1
  (32):          -1  2  1 -3  1
  (221):             1    -2  1
  (311):                1 -2  1
  (2111):                 -1  1
  (11111):                    1
For example, row 14 gives: h(32) = -e(32) + 2e(221) + e(311) - 3e(2111) + e(11111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

This is a regrouping of the triangle A321749. Row sums are A134286.

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