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

1, 1, 4, 9, 27, 60, 170, 377, 996, 2288, 5715, 13002, 32321, 72864, 175137, 400039, 943454, 2133159, 4993737, 11236889, 25995341, 58480330, 133650880, 299347432, 681346296, 1519116099, 3427954877, 7631479391, 17122129103, 37958987956, 84819325972, 187405201004

Offset: 0

Author

Alois P. Heinz, May 16 2018

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Ferrers Diagram
• Wikipedia, Domino
• Wikipedia, Domino tiling
• Wikipedia, Ferrers diagram
• Wikipedia, Partition (number theory)
• Wikipedia, Polyomino
• Wikipedia, Young tableau, Diagrams

Crossrefs

Cf. A304662, A304680.

Programs

Maple

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od; h(subsop(k=1, l), f)+
        `if`(nops(f)>0 and f[1]>=k, 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`(l=[], 1, h([0$l[1]], subsop(1=[][], l))):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
a:= n-> b(n$2, []):
seq(a(n), n=0..23);