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Number of triples (p,q,r) of partitions such that p is a partition of n and r <= q <= p (by diagram containment).
2

%I #10 May 24 2018 20:44:28

%S 1,3,12,34,100,246,630,1433,3298,7124,15283,31358,64100,126406,247587,

%T 472864,895548,1661690,3059734,5538991,9950980,17631398,31004004,

%U 53878023,92979904,158806852,269448833,453099946,757152246,1255180557,2068707378,3385065586

%N Number of triples (p,q,r) of partitions such that p is a partition of n and r <= q <= p (by diagram containment).

%H Alois P. Heinz, <a href="/A305023/b305023.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>

%e a(0) = 1: ((),(),()).

%e a(1) = 3: (1,(),()), (1,1,()), (1,1,1).

%e a(2) = 12: (11,(),()), (11,1,()), (11,1,1), (11,11,()), (11,11,1), (11,11,11), (2,(),()), (2,1,()), (2,1,1), (2,2,()), (2,2,1), (2,2,2).

%e a(3) = 34: (111,(),()), (111,1,()), (111,1,1), (111,11,()), (111,11,1), (111,11,11), (111,111,()), (111,111,1), (111,111,11), (111,111,111), (21,(),()), (21,1,()), (21,1,1), (21,11,()), (21,11,1), (21,11,11), (21,2,()), (21,2,1), (21,2,2), (21,21,()), (21,21,1), (21,21,11), (21,21,2), (21,21,21), (3,(),()), (3,1,()), (3,1,1), (3,2,()), (3,2,1), (3,2,2), (3,3,()), (3,3,1), (3,3,2), (3,3,3).

%Y Cf. A000041, A297388.

%K nonn

%O 0,2

%A _Alois P. Heinz_, May 23 2018