A321920 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in s(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and s is Schur functions.

1, -1, 1, 1, 0, 1, -2, 1, -1, 1, 0, 1, 0, 0, -1, 1, 2, -3, 1, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, -2, -2, 3, 3, -4, 1, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0

Offset

1,7

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

Crossrefs

Row sums are A134286. This is a regrouping of the triangle A321755.

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

Sequence in context: A210825 A056226 A044935 * A106276 A305053 A374081

Adjacent sequences: A321917 A321918 A321919 * A321921 A321922 A321923

Keyword

sign,tabf

Author

Gus Wiseman, Nov 22 2018

Status

approved