A321926 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in p(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and p is power sum symmetric functions.

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

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..80.

Wikipedia, Symmetric polynomial

Example

Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   1 -1
  (11):  1  1
.
  (3):    1 -1  1
  (21):   1    -1
  (111):  1  2  1
.
  (4):     1    -1  1 -1
  (22):    1  2 -1 -1  1
  (31):    1 -1        1
  (211):   1     1 -1 -1
  (1111):  1  2  3  3  1
.
  (5):      1 -1        1 -1  1
  (41):     1    -1  1       -1
  (32):     1 -1  1 -1     1 -1
  (221):    1     1  1 -2     1
  (311):    1  1 -1 -1     1  1
  (2111):   1  2  1 -1    -2 -1
  (11111):  1  4  5  5  6  4  1
For example, row 14 gives: p(32) = s(5) + s(32) - s(41) - s(221) + s(2111) - s(11111).

Crossrefs

Row sums are A317552. This is a regrouping of the triangle A321765.

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

Sequence in context: A355905 A368183 A367905 * A037870 A250205 A326017

Adjacent sequences: A321923 A321924 A321925 * A321927 A321928 A321929

Keyword

sign,tabf

Author

Gus Wiseman, Nov 22 2018

Status

approved