A067228 Number of rectangular standard Young tableaux with n cells.

1, 2, 2, 4, 2, 12, 2, 30, 44, 86, 2, 1190, 2, 860, 12014, 26886, 2, 184758, 2, 3359202, 2771342, 117574, 2, 327618902, 701149022, 1485802, 828630662, 27350160662, 2, 808310933492, 2, 2979826568702, 291724349282, 259289582, 557214344578322, 2031957220875002, 2

Offset: 1

Author

Naohiro Nomoto, Feb 20 2002

Keywords

• easy
• nonn

Comments

Number of ways to arrange the numbers 1, 2, .., n=i*j into an i*j rectangle so that each row and each column is increasing.

a(p) = 2 for prime p. - Alois P. Heinz, Jul 25 2012

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A000085, A060854, A067231.

Column k=0 of A238707.

Programs

Maple

with(numtheory):
a:= n-> n! * add(mul(k!/(i+k)!, k=0..(n/i)-1), i=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 25 2012

Mathematica

a[n_] := n! * Sum[Product[k!/(i+k)!, {k, 0, n/i-1}], {i, Divisors[n]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 02 2015, after Alois P. Heinz *)

Formula

a(n) = n! * Sum_{i|n} Product_{k=0..n/i-1} k!/(i+k)!. - Alois P. Heinz, Jul 25 2012

Extensions

Better name from Joerg Arndt, Feb 24 2014