A321761 -id:A321761 - cp's OEIS frontend

A321762 Sum of coefficients of monomial symmetric functions in the Schur function of the integer partition with Heinz number n.

1, 1, 2, 1, 3, 3, 5, 1, 4, 7, 7, 4, 11, 13, 12, 1, 15, 8, 22, 11, 30, 24, 30, 5, 14, 39, 9, 25, 42, 33, 56, 1, 59, 64, 47, 13, 77, 98, 113, 16, 101, 90, 135, 50, 43, 150, 176, 6, 53, 48, 195, 94, 231, 22, 119, 41, 331, 219, 297, 62, 385, 322, 141, 1, 250, 211

Offset: 1

Gus Wiseman, Nov 20 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sum of coefficients of s(41) = m(32) + m(41) + 2m(221) + 2m(311) + 3m(2111) + 4m(11111) is a(14) = 13.

Wikipedia, Symmetric polynomial

Row sums of A321761.

Cf. A000085, A008480, A056239, A124794, A124795, A153452, A296150, A296188, A300121, A319193, A321742-A321765.

A321924 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in s(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 2, 3, 0, 0, 0, 1, 3, 0, 0, 0, 0, 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, 1, 0, 2, 5, 0, 0, 0, 1, 1, 3, 6, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 22 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Tetrangle begins (zeroes not shown):
  (1): 1
.
  (2):  1 1
  (11):   1
.
  (3):   1 1 1
  (21):    1 2
  (111):     1
.
  (4):    1 1 1 1 1
  (22):     1   1 2
  (31):     1 1 2 3
  (211):        1 3
  (1111):         1
.
  (5):     1 1 1 1 1 1 1
  (41):      1 1 2 2 3 4
  (32):        1 2 1 3 5
  (221):         1   2 5
  (311):         1 1 3 6
  (2111):            1 4
  (11111):             1
For example, row 14 gives: s(32) = m(32) + 2m(221) + m(311) + 3m(2111) + 5m(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321761.

Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

cp's OEIS Frontend

A321762 Sum of coefficients of monomial symmetric functions in the Schur function of the integer partition with Heinz number n.

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