A321761 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in s(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, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 4, 0, 0, 1, 2, 1, 3, 5, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Nov 20 2018

Keywords

• nonn
• 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).

Examples

			Triangle begins:
   1
   1
   1   1
   0   1
   1   1   1
   0   1   2
   1   1   1   1   1
   0   0   1
   0   1   0   1   2
   0   1   1   2   3
   1   1   1   1   1   1   1
   0   0   0   1   3
   1   1   1   1   1   1   1   1   1   1   1
   0   1   1   2   2   3   4
   0   0   1   2   1   3   5
   0   0   0   0   1
   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
   0   0   0   1   0   2   5
For example, row 15 gives: s(32) = m(32) + 2m(221) + m(311) + 3m(2111) + 5m(11111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

Row sums are A321762.

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

Formula

If s(y) = Sum_{|z| = |y|} c(y,z) * m(z), then Sum_{|z| = |y|} c(y,z) * P(z) = A296188(H(y)), where P(y) is the number of distinct permutations of y.