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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.

%I A304789 #30 Feb 16 2025 08:33:54
%S A304789 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,
%T A304789 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,
%U A304789 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
%N 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.
%H A304789 Alois P. Heinz, <a href="/A304789/b304789.txt">Rows n = 0..20, flattened</a>
%H A304789 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H A304789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>
%H A304789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A304789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%H A304789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mutilated_chessboard_problem">Mutilated chessboard problem</a>
%H A304789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A304789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>
%H A304789 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%F A304789 Sum_{k>0} k * T(n,k) = A304662(n).
%F A304789 T(n,A304790(n)) = 1 for n in { A001105 }.
%F A304789 Sum_{k>=0} T(n,k) = A058696(n) = A000041(2n).
%F A304789 Sum_{k>=1} T(n,k) = A000712(n).
%F A304789 Sum_{k>=2} T(n,k) = A048574(n) = A052837(n).
%e A304789 T(2,2) = 1: 22.
%e A304789 T(3,0) = 1: 321.
%e A304789 T(3,1) = 6: 111111, 21111, 3111, 411, 51, 6.
%e A304789 T(3,2) = 2: 2211, 42.
%e A304789 T(3,3) = 2: 222, 33.
%e A304789 T(8,36) = 1: 4444.
%e A304789 Triangle T(n,k) begins:
%e A304789    0,  1;
%e A304789    0,  2;
%e A304789    0,  4, 1;
%e A304789    1,  6, 2,  2;
%e A304789    2, 10, 3,  4,  1,  2;
%e A304789    6, 14, 4,  6,  4,  4, 0, 2, 2;
%e A304789   12, 22, 5,  8,  7,  6, 2, 4, 4, 0, 0, 4, 1, 2;
%e A304789   25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2;
%p A304789 h:= proc(l, f) option remember; local k; if min(l[])>0 then
%p A304789      `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
%p A304789     else for k from nops(l) while l[k]>0 by -1 do od;
%p A304789         `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
%p A304789         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
%p A304789       fi
%p A304789     end:
%p A304789 g:= l-> x^`if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
%p A304789           `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
%p A304789 b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
%p A304789                   +b(n-i, min(n-i, i), [l[], i])):
%p A304789 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n$2, [])):
%p A304789 seq(T(n), n=0..11);
%Y A304789 Columns k=0-1 give: A304710, A139582(n) = 2*A000041(n) for n>0.
%Y A304789 Row sums give A058696(n) or A000041(2n).
%Y A304789 Cf. A000290, A000712, A001105, A048574, A052837, A304662, A304790.
%K A304789 nonn,look,tabf
%O A304789 0,4
%A A304789 _Alois P. Heinz_, May 18 2018