A321886 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in f(u), where H is Heinz number, m is monomial symmetric functions, and f is forgotten symmetric functions.

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

Offset

1,10

Comments

Row n has length A000041(A056239(n)).

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

a(n) is also the coefficient of f(v) in m(u).

Links

Table of n, a(n) for n=1..83.

Wikipedia, Symmetric polynomial

Example

Triangle begins:
   1
   1
  -1   0
   1   1
   1   0   0
  -2  -1   0
  -1   0   0   0   0
   1   1   1
   1   1   0   0   0
   2   0   1   0   0
   1   0   0   0   0   0   0
  -3  -2  -2  -1   0
  -1   0   0   0   0   0   0   0   0   0   0
  -2  -1   0   0   0   0   0
  -2   0  -1   0   0   0   0
   1   1   1   1   1
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   3   1   2   1   0   0   0
For example, row 12 gives: f(211) = -3m(4) - 2m(22) - 2m(31) - m(211).

Crossrefs

Row sums are A321887.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A135278, A300121, A319193, A321742-A321765.

Sequence in context: A227840 A116852 A269245 * A060154 A061007 A060838

Adjacent sequences: A321883 A321884 A321885 * A321887 A321888 A321889

Keyword

sign,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved