A321764 -id:A321764 - cp's OEIS frontend

A321763 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and s is Schur functions.

1, 1, 1, -1, 0, 1, 1, -1, 1, 0, 1, -2, 1, 0, -1, 1, -1, 0, 0, 1, 0, 1, 0, -1, 1, 0, -1, 1, -1, 2, 1, -1, 0, 0, 1, -1, 1, 0, 0, 0, 1, -3

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).

			Triangle begins:
   1
   1
   1  -1
   0   1
   1  -1   1
   0   1  -2
   1   0  -1   1  -1
   0   0   1
   0   1   0  -1   1
   0  -1   1  -1   2
   1  -1   0   0   1  -1   1
   0   0   0   1  -3
For example, row 10 gives: m(31) = -s(22) + s(31) - s(211) + 2s(1111).

Wikipedia, Symmetric polynomial

Row sums are A321764.

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

A321894 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in f(u), where H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, -1, 1, 1, 0, 1, -1, 1, -2, 1, 0, -1, 0, 1, -1, 1, 1, 0, 0, 1, 1, -1, 0, 0, 2, -1, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, -3, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 0, 1, -1, 1, -2, 1, 1, -1, -1, 1, 0, -2, 2, -1, 1, -1, 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).

			Triangle begins:
   1
   1
  -1   1
   1   0
   1  -1   1
  -2   1   0
  -1   0   1  -1   1
   1   0   0
   1   1  -1   0   0
   2  -1  -1   1   0
   1  -1   0   0   1  -1   1
  -3   0   1   0   0
  -1   0   1   0   0  -1   0   0   1  -1   1
  -2   1   1  -1  -1   1   0
  -2   2  -1   1  -1   0   0
For example, row 15 gives: f(32) = -2s(5) - s(32) + 2s(41) + s(221) - s(311).

Wikipedia, Symmetric polynomial

Row sums are A321764.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A300121, A317552, A321742-A321765, A321892.

cp's OEIS Frontend

A321763 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and s is Schur functions.

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