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Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in s(u), where H is Heinz number, m is monomial symmetric functions, and s is Schur functions.
2

%I #4 Nov 20 2018 19:45:53

%S 1,1,1,1,0,1,1,1,1,0,1,2,1,1,1,1,1,0,0,1,0,1,0,1,2,0,1,1,2,3,1,1,1,1,

%T 1,1,1,0,0,0,1,3,1,1,1,1,1,1,1,1,1,1,1,0,1,1,2,2,3,4,0,0,1,2,1,3,5,0,

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

%N Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in s(u), where H is Heinz number, m is monomial 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>

%F If s(y) = Sum_{|z| = |y|} c(y,z) * m(z), then Sum_{|z| = |y|} c(y,z) * P(z) = A296188(H(y)), where P(y) is the number of distinct permutations of y.

%e Triangle begins:

%e 1

%e 1

%e 1 1

%e 0 1

%e 1 1 1

%e 0 1 2

%e 1 1 1 1 1

%e 0 0 1

%e 0 1 0 1 2

%e 0 1 1 2 3

%e 1 1 1 1 1 1 1

%e 0 0 0 1 3

%e 1 1 1 1 1 1 1 1 1 1 1

%e 0 1 1 2 2 3 4

%e 0 0 1 2 1 3 5

%e 0 0 0 0 1

%e 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 0 0 0 1 0 2 5

%e For example, row 15 gives: s(32) = m(32) + 2m(221) + m(311) + 3m(2111) + 5m(11111).

%Y Row sums are A321762.

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

%K nonn,tabf

%O 1,12

%A _Gus Wiseman_, Nov 20 2018