login
Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.
9

%I #6 Nov 20 2018 16:30:31

%S 1,1,1,0,1,2,1,0,0,1,1,0,1,0,0,0,0,1,3,6,1,2,0,0,0,1,0,1,0,0,1,0,0,0,

%T 0,0,0,1,2,2,2,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,1,

%U 6,4,12,24,1,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum 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).

%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 2

%e 1 0 0

%e 1 1 0

%e 1 0 0 0 0

%e 1 3 6

%e 1 2 0 0 0

%e 1 0 1 0 0

%e 1 0 0 0 0 0 0

%e 1 2 2 2 0

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

%e 1 1 0 0 0 0 0

%e 1 0 1 0 0 0 0

%e 1 6 4 12 24

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

%e 1 1 2 2 0 0 0

%e For example, row 18 gives: p(221) = m(5) + 2m(32) + m(41) + 2m(221).

%Y Row sums are A321751.

%Y Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A321742-A321765.

%K nonn,tabf

%O 1,6

%A _Gus Wiseman_, Nov 20 2018