A304789 Number T(n,k) of partitions of 2n whose Ferrers-Young diagram allows exactly k different domino tilings; triangle T(n,k), n>=0, 0<=k<=A304790(n), read by rows.

0, 1, 0, 2, 0, 4, 1, 1, 6, 2, 2, 2, 10, 3, 4, 1, 2, 6, 14, 4, 6, 4, 4, 0, 2, 2, 12, 22, 5, 8, 7, 6, 2, 4, 4, 0, 0, 4, 1, 2, 25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2, 46, 44, 7, 12, 17, 14, 8, 8, 8, 0, 4, 12, 5, 6, 0, 8, 2, 0, 8, 4, 0, 4, 0, 0, 0, 2, 2, 0, 0, 4, 1, 2, 0, 0, 2, 0, 1

Offset: 0

Author

Alois P. Heinz, May 18 2018

Keywords

• nonn
• look
• tabf

Examples

			T(2,2) = 1: 22.
T(3,0) = 1: 321.
T(3,1) = 6: 111111, 21111, 3111, 411, 51, 6.
T(3,2) = 2: 2211, 42.
T(3,3) = 2: 222, 33.
T(8,36) = 1: 4444.
Triangle T(n,k) begins:
   0,  1;
   0,  2;
   0,  4, 1;
   1,  6, 2,  2;
   2, 10, 3,  4,  1,  2;
   6, 14, 4,  6,  4,  4, 0, 2, 2;
  12, 22, 5,  8,  7,  6, 2, 4, 4, 0, 0, 4, 1, 2;
  25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2;
		

Links

• Alois P. Heinz, Rows n = 0..20, flattened
• 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, Young tableau, Diagrams
• Index entries for sequences related to dominoes

Crossrefs

Columns k=0-1 give: A304710, A139582(n) = 2*A000041(n) for n>0.

Row sums give A058696(n) or A000041(2n).

Cf. A000290, A000712, A001105, A048574, A052837, A304662, A304790.

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;
        `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-> x^`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])):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n$2, [])):
seq(T(n), n=0..11);

Formula

Sum_{k>0} k * T(n,k) = A304662(n).

T(n,A304790(n)) = 1 for n in { A001105 }.

Sum_{k>=0} T(n,k) = A058696(n) = A000041(2n).

Sum_{k>=1} T(n,k) = A000712(n).

Sum_{k>=2} T(n,k) = A048574(n) = A052837(n).