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

0,4

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

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).

Example

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;

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);

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.

Sequence in context: A188448 A370412 A355917 * A264379 A090888 A154794

Adjacent sequences: A304786 A304787 A304788 * A304790 A304791 A304792

Keyword

nonn,look,tabf

Author

Alois P. Heinz, May 18 2018

Status

approved