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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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: A291929 A327807 A188448 * A264379 A090888 A154794

Adjacent sequences:  A304786 A304787 A304788 * A304790 A304791 A304792

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, May 18 2018

STATUS

approved

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Last modified October 16 14:39 EDT 2019. Contains 328096 sequences. (Running on oeis4.)