A321922 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in s(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and s is Schur functions.

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

Offset

1,13

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..81.

Wikipedia, Symmetric polynomial

Example

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

Crossrefs

Row sums are A155972. This is a regrouping of the triangle A321758.

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

Sequence in context: A203905 A309142 A064532 * A362496 A360003 A287146

Adjacent sequences: A321919 A321920 A321921 * A321923 A321924 A321925

Keyword

sign,tabf

Author

Gus Wiseman, Nov 22 2018

Status

approved