A321935 Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.

1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 2, -3, 1, 6, 3, 8, 6, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 2, -1, 0, -2, 1, -6, 3, 8, -6, 1, 24, 30, 20, 15, 20, 10, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, 0, 6, -4, 3, -4, -2, 1, 4, 0, 0, -5, 0, 0, 1, -6, 0, 5, 0, 5

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).

We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is the basis of Schur functions and syt(y) is the number of standard Young tableaux of shape y.

Examples

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):   1  1
  (11): -1  1
.
  (3):    2  3  1
  (21):  -1     1
  (111):  2 -3  1
.
  (4):     6  3  8  6  1
  (22):       3 -4     1
  (31):   -2 -1     2  1
  (211):   2 -1    -2  1
  (1111): -6  3  8 -6  1
.
  (5):     24 30 20 15 20 10  1
  (41):    -6    -5     5  5  1
  (32):       -6  4  3 -4  2  1
  (221):       6 -4  3 -4 -2  1
  (311):    4       -5        1
  (2111):  -6     5     5 -5  1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

This is a regrouping of the triangle A321900.

Cf. A008480, A056239, A124794, A124795, A153452 (standard Young tableaux), A215366, A296188, A300121, A319191, A319193, A321908, A321912-A321934.