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

1, 1, 2, 6, 6, 23, 16, 76, 42, 239, 106, 688, 268, 1931, 650, 5266, 1580, 13861, 3750, 35810, 8862, 91065, 20598, 226914, 47776, 559271, 109248, 1360152, 248966, 3270429, 562630, 7785974, 1264780, 18378067, 2823958, 43007532, 6282198, 99892837, 13884820

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
• Index entries for sequences related to dominoes

Crossrefs

Bisection (even part) gives A304662.

Cf. A304677.

Programs

Maple

h:= proc(l, f, t) 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), t))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(t, h(subsop(k=1, l), f, false), 0)+
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f, t), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f, t), 0)
      fi
    end:
g:= l-> (t-> `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l),
               is(t, odd))))(add(i, i=l)):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$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);