A296625 -id:A296625 - cp's OEIS frontend

A304662 Total number of domino tilings of Ferrers-Young diagrams summed over all partitions of 2n.

1, 2, 6, 16, 42, 106, 268, 650, 1580, 3750, 8862, 20598, 47776, 109248, 248966, 562630, 1264780, 2823958, 6282198, 13884820, 30590124, 67051982, 146463790, 318588916, 690882926, 1492592450, 3215372064, 6904561416, 14786529836, 31574656096, 67261524262

Offset: 0

Alois P. Heinz, May 16 2018

nonn

			a(2) = 6:
._.  .___.  ._._.  .___.  ._.___.  .___.___.
| |  |___|  | | |  |___|  | |___|  |___|___|
|_|  | |    |_|_|  |___|  |_|
| |  |_|
|_|

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
Gus Wiseman, All 42 domino tilings of integer partitions of 8.
Gus Wiseman, All 106 domino tilings of integer partitions of 10.
Index entries for sequences related to dominoes

Row sums of A304718.
Bisection (even part) of A304680.
Cf. A004003, A296625, A300060, A300789, A304677, A304710, A304790.

h:= proc(l, f) 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)))
    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-> `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])):
a:= n-> b(2*n$2, []):
seq(a(n), n=0..12);

h[l_, f_] := h[l, f] = Module[{k}, If[Min[l]>0, If[Length[f] == 0, 1, h[l[[1 ;; f[[1]]]]-1, ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]]>0, k--]; If[Length[f]>0 && f[[1]] >= k, h[ReplacePart[l, k -> 2], f], 0] + If[k>1 && l[[k-1]] == 0, h[ReplacePart[l, {k -> 1, k-1 -> 1}], f], 0]]];
g[l_] := If[Sum[If[OddQ[l[[i]]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, {l[[1]]}], ReplacePart[l, 1 -> Nothing]]], 0];
b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]], b[n, i-1, l] + b[n-i, Min[n-i, i], Append[l, i]]];
a[n_] := b[2n, 2n, {}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 29 2021, after Alois P. Heinz *)

a(n) = Sum_{k=0..A304790(n)} k * A304789(n,k).
a(n) = Sum_{k=0..n} A304718(n,k).
a(n) = A296625(n) for n < 7.

A296624 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(mu) with partition lambda >= mu and size(lambda) + size(mu)= n.

Original entry on oeis.org

1, 1, 4, 7, 20, 37, 90, 171, 378, 721, 1500, 2843, 5682, 10661, 20674

Offset: 0

Wouter Meeussen, Dec 17 2017

The condition lambda >= mu restricts the results to the lower triangular part of the matrix formed by products of all pairs of partitions.

'Multiplicity' signifies that terms like k*s(nu) count as k terms.

			For n=3 we have
s(3)*s(0) = s(3); s(2,1)*s(0) = s(2,1); s(1,1,1)*s(0) = s(1,1,1)
s(2)*s(1) = s(3) + s(2,1) and
s(1,1)*s(1) = s(2,1) + s(1,1,1)
for a total of 3+2+2 = 7 terms.

Wouter Meeussen, Mathematica toolbox for symmetric functions

Cf. A296625, A296626.

Tr/@ Table[Sum[
  Length[LRRule[\[Lambda], \[Mu]]], {\[Lambda],
   Partitions[n - i]}, {\[Mu],
   If[2 i === n, Join[{\[Lambda]}, lesspartitions[\[Lambda]]],
    Partitions[i]]}], {n, 14}, {i, 0, Floor[(n)/2]}]; (* Uses functions defined in the 'Toolbox for symmetric functions', see Links. *)

A296626 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(mu) with size(lambda) + size(mu) = n.

Original entry on oeis.org

1, 2, 6, 14, 34, 74, 164, 342, 714, 1442, 2894, 5686, 11096, 21322, 40688

Offset: 0

Wouter Meeussen, Dec 17 2017

Equals 2*A296624 - aerated version of A296625, so s(lambda)*s(mu) is counted again as s(mu)*s(lambda) if mu <> lambda. The aerated version of A296625 reads as 1, 0, 2, 0, 6, 0, 16, 0, 42, 0, 106, 0, ...

For a program see A296624 and A296625.

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A304662 Total number of domino tilings of Ferrers-Young diagrams summed over all partitions of 2n.

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A296624 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(mu) with partition lambda >= mu and size(lambda) + size(mu)= n.

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Mathematica

A296626 a(n) is the total multiplicity of all products of Schur functions s(lambda)*s(mu) with size(lambda) + size(mu) = n.

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