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A067231
Number of Young tableaux with n=i*j cells and type i*j matrices with i>=j.
2
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
1,4
COMMENTS
a(p) = 1 for prime p. - Alois P. Heinz, Jul 25 2012
LINKS
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
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Naohiro Nomoto, Feb 20 2002
STATUS
approved