A321758 - OEIS

%I #4 Nov 20 2018 16:31:31

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

%T 0,0,0,0,0,1,-1,-1,1,0,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,1,0,

%U 0,0,0,-1,1,2,-3,1,1,0,0,0,0,0,0,0,0,0,0

%N Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in s(u), where H is Heinz number, h is homogeneous 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   0

%e   -1   1

%e    1   0   0

%e   -1   1   0

%e    1   0   0   0   0

%e    1  -2   1

%e    0   1  -1   0   0

%e   -1   0   1   0   0

%e    1   0   0   0   0   0   0

%e    1  -1  -1   1   0

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

%e   -1   1   0   0   0   0   0

%e    0  -1   1   0   0   0   0

%e   -1   1   2  -3   1

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

%e    0   1  -1   1  -1   0   0

%e For example, row 18 gives: s(221) = -h(32) + h(41) + h(221) - h(311).

%Y Row sums are A010051.

%Y Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A300121, A304438, A317552, A317554, A321742-A321765.

%K sign,tabf

%O 1,19

%A _Gus Wiseman_, Nov 20 2018