A321746 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and e is elementary symmetric functions.

1, 1, -2, 1, 1, 0, 3, -3, 1, -3, 1, 0, -4, 2, 4, -4, 1, 1, 0, 0, 2, 1, -2, 0, 0, 4, -2, -1, 1, 0, 5, -5, -5, 5, 5, -5, 1, -4, 0, 1, 0, 0, -6, 6, 6, 3, -2, -6, -12, 9, 6, -6, 1, -5, 1, 5, -3, -1, 1, 0, -5, 5, -1, 1, -2, 0, 0, 1, 0, 0, 0, 0, 7, -7, -7, -7, 14, 7

Offset: 1

Author

Gus Wiseman, Nov 19 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).

Also the coefficient of h(v) in f(u), where h is homogeneous symmetric functions and f is forgotten symmetric functions.

Examples

			Triangle begins:
   1
   1
  -2   1
   1   0
   3  -3   1
  -3   1   0
  -4   2   4  -4   1
   1   0   0
   2   1  -2   0   0
   4  -2  -1   1   0
   5  -5  -5   5   5  -5   1
  -4   0   1   0   0
  -6   6   6   3  -2  -6 -12   9   6  -6   1
  -5   1   5  -3  -1   1   0
  -5   5  -1   1  -2   0   0
   1   0   0   0   0
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   5  -3   1   0   0   0   0
For example, row 10 gives: m(31) = 4e(4) - 2e(22) - e(31) + e(211).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are A321747.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A321738, A321742-A321765, A321854.