A321913 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in h(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 6, 1, 1, 1, 1, 1, 1, 3, 2, 4, 6, 1, 2, 2, 3, 4, 1, 4, 3, 7, 12, 1, 6, 4, 12, 24, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 1, 2, 3, 5, 4, 7, 10, 1, 3, 5, 11, 8, 18, 30, 1, 3, 4, 8, 7, 13, 20, 1, 4, 7, 18, 13, 33, 60, 1, 5

Offset: 1

Author

Gus Wiseman, Nov 22 2018

Keywords

• nonn
• tabf

Comments

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

Also the coefficient of f(v) in e(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

Examples

			Tetrangle begins:
  (1):  1
.
  (2):   1  1
  (11):  1  2
.
  (3):    1  1  1
  (21):   1  2  3
  (111):  1  3  6
.
  (4):     1  1  1  1  1
  (22):    1  3  2  4  6
  (31):    1  2  2  3  4
  (211):   1  4  3  7 12
  (1111):  1  6  4 12 24
.
  (5):      1  1  1  1  1  1  1
  (41):     1  2  2  3  3  4  5
  (32):     1  2  3  5  4  7 10
  (221):    1  3  5 11  8 18 30
  (311):    1  3  4  8  7 13 20
  (2111):   1  4  7 18 13 33 60
  (11111):  1  5 10 30 20 60 20
For example, row 14 gives: h(32) = m(5) + 3m(32) + 2m(41) + 5m(221) + 4m(311) + 7m(2111) + 10m(11111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

This is a regrouping of the triangle A321744.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.