A321925 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in m(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and m is monomial symmetric functions.

1, 1, -1, 0, 1, 1, -1, 1, 0, 1, -2, 0, 0, 1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 1, 0, -1, 1, -1, 2, 0, 0, 0, 1, -3, 0, 0, 0, 0, 1, 1, -1, 0, 0, 1, -1, 1, 0, 1, -1, 1, -1, 1, -2, 0, 0, 1, -1, -1, 2, -2, 0, 0, 0, 1, 0, -2, 3, 0, 0, 0, -1, 1, -1, 3, 0, 0, 0, 0, 0, 1

Offset

1,11

Comments

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..80.

Wikipedia, Symmetric polynomial

Example

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

Crossrefs

This is a regrouping of the triangle A321763.

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

Sequence in context: A335877 A125203 A023565 * A025922 A342322 A161369

Adjacent sequences: A321922 A321923 A321924 * A321926 A321927 A321928

Keyword

sign,tabf

Author

Gus Wiseman, Nov 22 2018

Status

approved