OFFSET
0,5
COMMENTS
Also the number of partitions of 2n where the number of odd parts in even positions differs from the number of odd parts in odd positions.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
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
EXAMPLE
a(3) = 1: the Ferrers-Young diagram of 321 cannot be tiled with dominoes because the numbers of white and black squares (when colored like a chessboard) are different but each domino covers exactly one white and one black square:
._____.
|_|X|_|
|X|_|
|_|
.
a(4) = 2: 32111, 521.
a(5) = 6: 3211111, 32221, 4321, 52111, 541, 721.
a(6) = 12: 321111111, 3222111, 33321, 432111, 5211111, 52221, 54111, 543, 6321, 72111, 741, 921.
MAPLE
b:= proc(n, i, p, c) option remember; `if`(n=0, `if`(c=0, 0, 1),
`if`(i<1, 0, b(n, i-1, p, c)+b(n-i, min(n-i, i), -p, c+
`if`(i::odd, p, 0))))
end:
a:= n-> b(2*n$2, 1, 0):
seq(a(n), n=0..50);
# second Maple program:
a:= n-> (p-> p(2*n)-add(p(j)*p(n-j), j=0..n))(combinat[numbpart]):
seq(a(n), n=0..50);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
numtheory[sigma](j)*b(n-j, k), j=1..n)*k/n)
end:
a:= n-> b(2*n, 1)-b(n, 2):
seq(a(n), n=0..50);
MATHEMATICA
b[n_, i_, p_, c_] := b[n, i, p, c] = If[n == 0, If[c == 0, 0, 1], If[i < 1, 0, b[n, i - 1, p, c] + b[n - i, Min[n - i, i], -p, c + If[OddQ[i], p, 0]]]];
a[n_] := b[2n, 2n, 1, 0];
Table[a[n], {n, 0, 50}]
(* second program: *)
a[n_] := PartitionsP[2n] - Sum[PartitionsP[j]* PartitionsP[n - j], {j, 0, n}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 17 2018
STATUS
approved