A321755 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in s(u), where H is Heinz number, e is elementary symmetric functions, and s is Schur functions.

1, 1, -1, 1, 1, 0, 1, -2, 1, -1, 1, 0, -1, 1, 2, -3, 1, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, 1, -2, -2, 3, 3, -4, 1, -1, 0, 1, 0, 0, -1, 2, 2, 1, -1, -3, -6, 6, 4, -5, 1, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 1, -2, -2, -2, 6, 3, 3

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).

Examples

			Triangle begins:
   1
   1
  -1   1
   1   0
   1  -2   1
  -1   1   0
  -1   1   2  -3   1
   1   0   0
   0   1  -1   0   0
   1  -1  -1   1   0
   1  -2  -2   3   3  -4   1
  -1   0   1   0   0
  -1   2   2   1  -1  -3  -6   6   4  -5   1
  -1   1   2  -2  -1   1   0
   0   1  -1   1  -1   0   0
   1   0   0   0   0
   1  -2  -2  -2   6   3   3   3  -4  -4 -12  10   5  -6   1
   0  -1   1   0   0   0   0
For example, row 14 gives: s(41) = -e(5) + 2e(32) + e(41) - 2e(221) - e(311) + e(2111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are A036987.

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