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

1, 1, 2, 5, 14, 28, 62, 150, 380, 787, 1760, 3951, 9338, 19536, 43224, 94326, 213278, 448193, 979712, 2094981, 4622262, 9670378, 20886560, 44067191, 95469402, 198712506

Offset: 0

Author

Alois P. Heinz, May 17 2018

Keywords

• nonn

Examples

			a(3) = 5:
:  .___.  ._.___.  .___.  ._._.  ._._.___.
:  |___|  | |___|  |___|  | | |  | | |___|
:  | |    |_|      | | |  |_|_|  |_|_|
:  |_|    | |      |_|_|  |___|
:  | |    |_|
:  |_|
		

Links

• Eric Weisstein's World of Mathematics, Ferrers Diagram
• Wikipedia, Domino
• Wikipedia, Domino tiling
• Wikipedia, Ferrers diagram
• Wikipedia, Mutilated chessboard problem
• Wikipedia, Partition (number theory)
• Wikipedia, Polyomino
• Wikipedia, Young tableau, Diagrams
• Index entries for sequences related to dominoes

Crossrefs

Cf. A304718.

Programs

Maple

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od; expand(
        `if`(nops(f)>0 and f[1]>=k, x*h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0))
      fi
    end:
g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
        `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
a:= n-> coeff(b(2*n$2, []), x, iquo(n, 2)):
seq(a(n), n=0..14);

Mathematica

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]]];
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];
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[n_] := CoefficientList[b[2 n, 2 n, {}], x];
a[n_] := T[n][[Floor[n/2] + 1]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz in A304718 *)

Formula

a(n) = A304718(n,floor(n/2)).