A321748 - OEIS

%I #4 Nov 20 2018 16:30:14

%S 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,

%T -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,

%U 9,5,-7,-9,9,-2,-5,5,11,-11,-8,10,-2,-1,1,2,-3,1,7

%N 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.

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

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

%C Also the coefficient of e(v) in f(u), where e is elementary symmetric functions and f is forgotten symmetric functions.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>

%e Triangle begins:

%e    1

%e    1

%e    2  -1

%e   -1   1

%e    3  -3   1

%e   -3   5  -2

%e    4  -2  -4   4  -1

%e    1  -2   1

%e   -2   3   2  -4   1

%e   -4   2   7  -7   2

%e    5  -5  -5   5   5  -5   1

%e    4  -4  -7  10  -3

%e    6  -6  -6  -3   2   6  12  -9  -6   6  -1

%e   -5   9   5  -7  -9   9  -2

%e   -5   5  11 -11  -8  10  -2

%e   -1   1   2  -3   1

%e    7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1

%e    5  -7 -11  14  10 -14   3

%e For example, row 10 gives: m(31) = -4h(4) + 2h(22) + 7h(31) - 7h(211) + 2h(1111).

%Y Row sums are A080339.

%Y Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A300121, A319191, A319193, A321738, A321742-A321765.

%K sign,tabf

%O 1,3

%A _Gus Wiseman_, Nov 20 2018