A321900 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in S(u), where H is Heinz number, p is power sum symmetric functions, and S is augmented Schur functions.

1, 1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 6, 3, 8, 6, 1, 2, -3, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 24, 30, 20, 15, 20, 10, 1, 2, -1, 0, -2, 1, 120, 90, 144, 40, 15, 90, 120, 45, 40, 15, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, -6, 3, 8, -6, 1, 720, 840

Offset: 1

Author

Gus Wiseman, Nov 20 2018

Keywords

• sign
• tabf

Comments

Row n has length A000041(A056239(n)).

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 Schur functions and syt(y) is the number of standard Young tableaux of shape y.

Examples

			Triangle begins:
    1
    1
    1    1
   -1    1
    2    3    1
   -1    0    1
    6    3    8    6    1
    2   -3    1
    0    3   -4    0    1
   -2   -1    0    2    1
   24   30   20   15   20   10    1
    2   -1    0   -2    1
  120   90  144   40   15   90  120   45   40   15    1
   -6    0   -5    0    5    5    1
    0   -6    4    3   -4    2    1
   -6    3    8   -6    1
  720  840  504  420  630  504  210  280  105  210  420  105   70   21    1
    0    6   -4    3   -4   -2    1
For example, row 15 gives: S(32) = 4p(32) - 6p(41) + 3p(221) - 4p(311) + 2p(2111) + p(11111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are above.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A296561, A300121, A304438, A317552, A317554, A321742-A321765.