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

1,12

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).

Links

Table of n, a(n) for n=1..87.

Wikipedia, Symmetric polynomial

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.

Example

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).

Crossrefs

Row sums are A321762.

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

Sequence in context: A316865 A325616 A056978 * A037819 A090405 A237837

Adjacent sequences: A321758 A321759 A321760 * A321762 A321763 A321764

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved