A321932 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in e(u) * Product_i u_i!, where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and e is elementary symmetric functions.

1, -1, 1, 0, 1, 2, -3, 1, 0, -1, 1, 0, 0, 1, -6, 3, 8, -6, 1, 0, 1, 0, -2, 1, 0, 0, 2, -3, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 24, -30, -20, 15, 20, -10, 1, 0, -6, 0, 3, 8, -6, 1, 0, 0, -2, 3, 2, -4, 1, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 2, -3, 1, 0, 0, 0, 0, 0

Offset

1,6

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..79.

Wikipedia, Symmetric polynomial

Example

Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):  -1  1
  (11):     1
.
  (3):    2 -3  1
  (21):     -1  1
  (111):        1
.
  (4):    -6  3  8 -6  1
  (22):       1    -2  1
  (31):          2 -3  1
  (211):           -1  1
  (1111):              1
.
  (5):     24 30 20 15 20 10  1
  (41):       -6     3  8 -6  1
  (32):          -2  3  2 -4  1
  (221):             1    -2  1
  (311):                2 -3  1
  (2111):                 -1  1
  (11111):                    1
For example, row 14 gives: 12e(32) = -2p(32) + 3p(221) + 2p(311) - 4p(2111) + p(11111).

Crossrefs

Row sums are A134286. This is a regrouping of the triangle A321896.

Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

Sequence in context: A124314 A059087 A353493 * A321933 A030373 A079343

Adjacent sequences: A321929 A321930 A321931 * A321933 A321934 A321935

Keyword

sign,tabf

Author

Gus Wiseman, Nov 23 2018

Status

approved