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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.
2

%I #5 Nov 20 2018 19:46:26

%S 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,

%T 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,

%U -1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0

%N 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.

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

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

%e Triangle begins:

%e 1

%e 1

%e -1 0

%e 1 1

%e 1 0 0

%e -2 -1 0

%e -1 0 0 0 0

%e 1 1 1

%e 1 1 0 0 0

%e 2 0 1 0 0

%e 1 0 0 0 0 0 0

%e -3 -2 -2 -1 0

%e -1 0 0 0 0 0 0 0 0 0 0

%e -2 -1 0 0 0 0 0

%e -2 0 -1 0 0 0 0

%e 1 1 1 1 1

%e 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

%e 3 1 2 1 0 0 0

%e For example, row 12 gives: f(211) = -3m(4) - 2m(22) - 2m(31) - m(211).

%Y Row sums are A321887.

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

%K sign,tabf

%O 1,10

%A _Gus Wiseman_, Nov 20 2018