A321748 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

1, 1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 4, -2, -4, 4, -1, 1, -2, 1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 5, -5, -5, 5, 5, -5, 1, 4, -4, -7, 10, -3, 6, -6, -6, -3, 2, 6, 12, -9, -6, 6, -1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, -1, 1, 2, -3, 1, 7

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).

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

Examples

			Triangle begins:
   1
   1
   2  -1
  -1   1
   3  -3   1
  -3   5  -2
   4  -2  -4   4  -1
   1  -2   1
  -2   3   2  -4   1
  -4   2   7  -7   2
   5  -5  -5   5   5  -5   1
   4  -4  -7  10  -3
   6  -6  -6  -3   2   6  12  -9  -6   6  -1
  -5   9   5  -7  -9   9  -2
  -5   5  11 -11  -8  10  -2
  -1   1   2  -3   1
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   5  -7 -11  14  10 -14   3
For example, row 10 gives: m(31) = -4h(4) + 2h(22) + 7h(31) - 7h(211) + 2h(1111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are A080339.

Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A300121, A319191, A319193, A321738, A321742-A321765.