A321756 -id:A321756 - cp's OEIS frontend

A321757 Sum of coefficients of Schur functions in the elementary symmetric function of the integer partition with Heinz number n.

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 5, 1, 2, 3, 10, 1, 7, 1, 5, 3, 2, 1, 13, 4, 2, 11, 5, 1, 8, 1, 26, 3, 2, 4, 20, 1, 2, 3, 14, 1, 8, 1, 5, 13, 2, 1, 38, 5, 10, 3, 5, 1, 32, 4, 14, 3, 2, 1, 23

Offset: 1

Gus Wiseman, Nov 20 2018

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

			The sum of coefficients of e(221) = s(32) + 2s(221) + s(311) + 2s(2111) + s(11111) is a(18) = 7.

Wikipedia, Symmetric polynomial

Row sums of A321756.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296150, A296188, A300121, A304438, A317552, A321742-A321765.

A321921 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in e(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and e is elementary symmetric functions.

Original entry on oeis.org

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

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
  (11): 1 1
.
  (3):       1
  (21):    1 1
  (111): 1 2 1
.
  (4):            1
  (22):     1   1 1
  (31):         1 1
  (211):    1 1 2 1
  (1111): 1 2 3 3 1
.
  (5):                 1
  (41):              1 1
  (32):          1   1 1
  (221):       1 2 1 2 1
  (311):         1 1 2 1
  (2111):    1 2 3 3 3 1
  (11111): 1 4 5 5 6 4 1
For example, row 14 gives: e(32) = s(221) + s(2111) + s(11111).

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321756.

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

cp's OEIS Frontend

A321757 Sum of coefficients of Schur functions in the elementary symmetric function of the integer partition with Heinz number n.

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