A321757 -id:A321757 - cp's OEIS frontend

A321756 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in e(u), where H is Heinz number, e is elementary symmetric functions, and s is Schur functions.

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

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
   0   1
   1   1
   0   0   1
   0   1   1
   0   0   0   0   1
   1   2   1
   0   1   0   1   1
   0   0   0   1   1
   0   0   0   0   0   0   1
   0   1   1   2   1
   0   0   0   0   0   0   0   0   0   0   1
   0   0   0   0   0   1   1
   0   0   0   1   0   1   1
   1   2   3   3   1
   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1
   0   0   1   2   1   2   1
For example, row 18 gives: e(221) = s(32) + 2s(221) + s(311) + 2s(2111) + s(11111).

Wikipedia, Symmetric polynomial

Row sums are A321757.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A300121, A304438, A317552, A317554, A319225, A319226, A321742-A321765.

A321759 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in h(u), where H is Heinz number, h is homogeneous symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 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   0
   1   1
   1   0   0
   1   1   0
   1   0   0   0   0
   1   2   1
   1   1   1   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
   1   1   2   1   0
For example, row 12 gives: h(211) = s(4) + s(22) + 2s(31) + s(211).

Wikipedia, Symmetric polynomial

Row sums are A321757.

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

cp's OEIS Frontend

A321756 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in e(u), where H is Heinz number, e is elementary symmetric functions, and s is Schur functions.

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