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

0,3

Links

Table of n, a(n) for n=0..25.

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A304718.

Sequence in context: A194124 A349094 A212340 * A022630 A376325 A047133

Adjacent sequences: A304716 A304717 A304718 * A304720 A304721 A304722

Keyword

nonn

Author

Alois P. Heinz, May 17 2018

Status

approved