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A304677 Total number of tilings of Ferrers-Young diagrams using dominoes and monominoes summed over all partitions of n.

%I A304677 #9 Feb 16 2025 08:33:54
%S A304677 1,1,4,9,27,60,170,377,996,2288,5715,13002,32321,72864,175137,400039,
%T A304677 943454,2133159,4993737,11236889,25995341,58480330,133650880,
%U A304677 299347432,681346296,1519116099,3427954877,7631479391,17122129103,37958987956,84819325972,187405201004
%N A304677 Total number of tilings of Ferrers-Young diagrams using dominoes and monominoes summed over all partitions of n.
%H A304677 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H A304677 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>
%H A304677 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A304677 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%H A304677 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A304677 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyomino">Polyomino</a>
%H A304677 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>
%p A304677 h:= proc(l, f) option remember; local k; if min(l[])>0 then
%p A304677      `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
%p A304677     else for k from nops(l) while l[k]>0 by -1 do od; h(subsop(k=1, l), f)+
%p A304677         `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
%p A304677         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
%p A304677       fi
%p A304677     end:
%p A304677 g:= l-> `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))):
%p A304677 b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
%p A304677                   +b(n-i, min(n-i, i), [l[], i])):
%p A304677 a:= n-> b(n$2, []):
%p A304677 seq(a(n), n=0..23);
%Y A304677 Cf. A304662, A304680.
%K A304677 nonn
%O A304677 0,3
%A A304677 _Alois P. Heinz_, May 16 2018