A067231 Number of Young tableaux with n=i*j cells and type i*j matrices with i>=j.

1, 1, 1, 3, 1, 6, 1, 15, 43, 43, 1, 595, 1, 430, 6007, 25455, 1, 92379, 1, 1679601, 1385671, 58787, 1, 163809451, 701149021, 742901, 414315331, 13675080331, 1, 404155466746, 1, 1489913284351, 145862174641, 129644791, 278607172289161, 1851800127304981, 1

Offset: 1

Author

Naohiro Nomoto, Feb 20 2002

Keywords

• easy
• nonn

Comments

a(p) = 1 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, A067228.

Programs

Maple

with(numtheory):
a:= n-> n!*add(mul(k!/(i+k)!, k=0..n/i-1),
        i=select(d-> is(d>=sqrt(n)), 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, Select[Divisors[n], # >= Sqrt[n]&]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Formula

a(n) = number of ways to arrange the numbers 1, 2, .., n=i*j in i*j matrices so that each row and each column is increasing. Here i and j satisfy i >= j.

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