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A304719 Number of domino tilings of Ferrers-Young diagrams of partitions of 2n using exactly floor(n/2) horizontally oriented dominoes.

%I A304719 #19 Feb 16 2025 08:33:54
%S A304719 1,1,2,5,14,28,62,150,380,787,1760,3951,9338,19536,43224,94326,213278,
%T A304719 448193,979712,2094981,4622262,9670378,20886560,44067191,95469402,
%U A304719 198712506
%N A304719 Number of domino tilings of Ferrers-Young diagrams of partitions of 2n using exactly floor(n/2) horizontally oriented dominoes.
%H A304719 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H A304719 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>
%H A304719 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A304719 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%H A304719 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mutilated_chessboard_problem">Mutilated chessboard problem</a>
%H A304719 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A304719 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyomino">Polyomino</a>
%H A304719 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>
%H A304719 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%F A304719 a(n) = A304718(n,floor(n/2)).
%e A304719 a(3) = 5:
%e A304719 :  .___.  ._.___.  .___.  ._._.  ._._.___.
%e A304719 :  |___|  | |___|  |___|  | | |  | | |___|
%e A304719 :  | |    |_|      | | |  |_|_|  |_|_|
%e A304719 :  |_|    | |      |_|_|  |___|
%e A304719 :  | |    |_|
%e A304719 :  |_|
%p A304719 h:= proc(l, f) option remember; local k; if min(l[])>0 then
%p A304719      `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
%p A304719     else for k from nops(l) while l[k]>0 by -1 do od; expand(
%p A304719         `if`(nops(f)>0 and f[1]>=k, x*h(subsop(k=2, l), f), 0)+
%p A304719         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0))
%p A304719       fi
%p A304719     end:
%p A304719 g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
%p A304719         `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
%p A304719 b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
%p A304719                   +b(n-i, min(n-i, i), [l[], i])):
%p A304719 a:= n-> coeff(b(2*n$2, []), x, iquo(n, 2)):
%p A304719 seq(a(n), n=0..14);
%t A304719 h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[l[[1 ;; f[[1]]]] - 1, ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]]>0, k--]; If[Length[f] > 0 && f[[1]] >= k, x*h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k - 1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]];
%t A304719 g[l_] := If[Sum[If[OddQ[l[[i]]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, {l[[1]]}], ReplacePart[l, 1 -> Nothing]]], 0];
%t A304719 b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]], b[n, i-1, l] + b[n-i, Min[n-i, i], Append[l, i]]];
%t A304719 T[n_] := CoefficientList[b[2 n, 2 n, {}], x];
%t A304719 a[n_] := T[n][[Floor[n/2] + 1]];
%t A304719 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Dec 28 2022, after _Alois P. Heinz_ in A304718 *)
%Y A304719 Cf. A304718.
%K A304719 nonn
%O A304719 0,3
%A A304719 _Alois P. Heinz_, May 17 2018