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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.

%I A321748 #4 Nov 20 2018 16:30:14
%S A321748 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 A321748 -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 A321748 9,5,-7,-9,9,-2,-5,5,11,-11,-8,10,-2,-1,1,2,-3,1,7
%N 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.
%C A321748 Row n has length A000041(A056239(n)).
%C A321748 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321748 Also the coefficient of e(v) in f(u), where e is elementary symmetric functions and f is forgotten symmetric functions.
%H A321748 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e A321748 Triangle begins:
%e A321748    1
%e A321748    1
%e A321748    2  -1
%e A321748   -1   1
%e A321748    3  -3   1
%e A321748   -3   5  -2
%e A321748    4  -2  -4   4  -1
%e A321748    1  -2   1
%e A321748   -2   3   2  -4   1
%e A321748   -4   2   7  -7   2
%e A321748    5  -5  -5   5   5  -5   1
%e A321748    4  -4  -7  10  -3
%e A321748    6  -6  -6  -3   2   6  12  -9  -6   6  -1
%e A321748   -5   9   5  -7  -9   9  -2
%e A321748   -5   5  11 -11  -8  10  -2
%e A321748   -1   1   2  -3   1
%e A321748    7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
%e A321748    5  -7 -11  14  10 -14   3
%e A321748 For example, row 10 gives: m(31) = -4h(4) + 2h(22) + 7h(31) - 7h(211) + 2h(1111).
%Y A321748 Row sums are A080339.
%Y A321748 Cf. A005651, A008480, A048994, A056239, A124794, A124795, A135278, A300121, A319191, A319193, A321738, A321742-A321765.
%K A321748 sign,tabf
%O A321748 1,3
%A A321748 _Gus Wiseman_, Nov 20 2018