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%I #8 May 28 2019 19:32:31
%S 1,3,8,24,71,226,718,7860,26669,91152,316194,1103506,3892806,13803606,
%T 43946652
%N Number of wide partitions whose first part is of size n.
%C A wide partition is a partition with the property that any sub-partition (meaning, a partition obtained by taking some of the parts) dominates its conjugate.
%C A special case of Rota's Basis Conjecture is a generalization of the Dinitz Conjecture, namely that there is a diagram - a Young Tableaux such that you see 1 through n in each row of size n and at most one of each digit in each column - if and only if the partition is wide.
%H T. Chow, C. Fan, M. Goemans, J. Vondrak, <a href="https://doi.org/10.1016/S0196-8858(03)00015-0">Wide Partitions, Latin Tableaux and Rota's Basis Conjecture</a>, Advances in Applied Mathematics, Vol. 31 (2003), No. 2, pp. 334-358.
%e If a wide partition has its first part of size n, then it has to fit in an n X n grid, or it itself does not dominate its conjugate. a(2) is equal to 3 because {2}, {2,1} and {2,2} are all wide partitions.
%K hard,more,nonn
%O 1,2
%A _Paul Raff_, Jan 02 2008
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