login
A321748
Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in m(u), where H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.
1
1, 1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 4, -2, -4, 4, -1, 1, -2, 1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 5, -5, -5, 5, 5, -5, 1, 4, -4, -7, 10, -3, 6, -6, -6, -3, 2, 6, 12, -9, -6, 6, -1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, -1, 1, 2, -3, 1, 7
OFFSET
1,3
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).
Also the coefficient of e(v) in f(u), where e is elementary symmetric functions and f is forgotten symmetric functions.
EXAMPLE
Triangle begins:
1
1
2 -1
-1 1
3 -3 1
-3 5 -2
4 -2 -4 4 -1
1 -2 1
-2 3 2 -4 1
-4 2 7 -7 2
5 -5 -5 5 5 -5 1
4 -4 -7 10 -3
6 -6 -6 -3 2 6 12 -9 -6 6 -1
-5 9 5 -7 -9 9 -2
-5 5 11 -11 -8 10 -2
-1 1 2 -3 1
7 -7 -7 -7 14 7 7 7 -7 -7 -21 14 7 -7 1
5 -7 -11 14 10 -14 3
For example, row 10 gives: m(31) = -4h(4) + 2h(22) + 7h(31) - 7h(211) + 2h(1111).
KEYWORD
sign,tabf
AUTHOR
Gus Wiseman, Nov 20 2018
STATUS
approved