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A304680 Total number of tilings of Ferrers-Young diagrams using dominoes and at most one monomino summed over all partitions of n.

%I A304680 #15 Feb 16 2025 08:33:54
%S A304680 1,1,2,6,6,23,16,76,42,239,106,688,268,1931,650,5266,1580,13861,3750,
%T A304680 35810,8862,91065,20598,226914,47776,559271,109248,1360152,248966,
%U A304680 3270429,562630,7785974,1264780,18378067,2823958,43007532,6282198,99892837,13884820
%N A304680 Total number of tilings of Ferrers-Young diagrams using dominoes and at most one monomino summed over all partitions of n.
%H A304680 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H A304680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>
%H A304680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A304680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%H A304680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A304680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyomino">Polyomino</a>
%H A304680 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>
%H A304680 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%p A304680 h:= proc(l, f, t) option remember; local k; if min(l[])>0 then
%p A304680      `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f), t))
%p A304680     else for k from nops(l) while l[k]>0 by -1 do od;
%p A304680         `if`(t, h(subsop(k=1, l), f, false), 0)+
%p A304680         `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f, t), 0)+
%p A304680         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f, t), 0)
%p A304680       fi
%p A304680     end:
%p A304680 g:= l-> (t-> `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l),
%p A304680                is(t, odd))))(add(i, i=l)):
%p A304680 b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
%p A304680                   +b(n-i, min(n-i, i), [l[], i])):
%p A304680 a:= n-> b(n$2, []):
%p A304680 seq(a(n), n=0..23);
%Y A304680 Bisection (even part) gives A304662.
%Y A304680 Cf. A304677.
%K A304680 nonn
%O A304680 0,3
%A A304680 _Alois P. Heinz_, May 16 2018