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Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in f(u), where H is Heinz number, f is forgotten symmetric functions, and s is Schur functions.
1

%I #5 Nov 21 2018 09:24:31

%S 1,1,-1,1,1,0,1,-1,1,-2,1,0,-1,0,1,-1,1,1,0,0,1,1,-1,0,0,2,-1,-1,1,0,

%T 1,-1,0,0,1,-1,1,-3,0,1,0,0,-1,0,1,0,0,-1,0,0,1,-1,1,-2,1,1,-1,-1,1,0,

%U -2,2,-1,1,-1,0,0

%N Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in f(u), where H is Heinz number, f is forgotten symmetric functions, and s is Schur 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).

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

%e Triangle begins:

%e 1

%e 1

%e -1 1

%e 1 0

%e 1 -1 1

%e -2 1 0

%e -1 0 1 -1 1

%e 1 0 0

%e 1 1 -1 0 0

%e 2 -1 -1 1 0

%e 1 -1 0 0 1 -1 1

%e -3 0 1 0 0

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

%e -2 1 1 -1 -1 1 0

%e -2 2 -1 1 -1 0 0

%e For example, row 15 gives: f(32) = -2s(5) - s(32) + 2s(41) + s(221) - s(311).

%Y Row sums are A321764.

%Y Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A300121, A317552, A321742-A321765, A321892.

%K sign,tabf,more

%O 1,10

%A _Gus Wiseman_, Nov 20 2018