A321889 -id:A321889 - cp's OEIS frontend

A321888 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of f(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and f is forgotten symmetric functions.

1, 1, -1, 0, 1, 2, 1, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 1, 3, 6, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, -2, -2, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 20 2018

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).
Up to sign, a(n) is also the coefficient of m(v) in p(u), where m is monomial symmetric functions.

			Triangle begins:
   1
   1
  -1   0
   1   2
   1   0   0
  -1  -1   0
  -1   0   0   0   0
   1   3   6
   1   2   0   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
  -1  -2  -2  -2   0
  -1   0   0   0   0   0   0   0   0   0   0
  -1  -1   0   0   0   0   0
  -1   0  -1   0   0   0   0
   1   6   4  12  24
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   1   1   2   2   0   0   0
For example, row 12 gives: p(211) = -f(4) - 2f(22) - 2f(31) - 2f(211).

Wikipedia, Symmetric polynomial

Row sums are A321889.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319193, A321742-A321765.

cp's OEIS Frontend

A321888 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of f(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and f is forgotten symmetric functions.

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