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URL: https://oeis.org/A019576

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A019576
Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives triangle of numbers f(n,k)/n.
7
1, 1, 1, 2, 6, 1, 6, 45, 12, 1, 24, 420, 160, 20, 1, 120, 4800, 2450, 375, 30, 1, 720, 65520, 43050, 7560, 756, 42, 1, 5040, 1045170, 858480, 167825, 19208, 1372, 56, 1, 40320, 19126800, 19208000, 4110120, 516096, 43008, 2304, 72, 1
OFFSET
1,4
COMMENTS
T(n,k) is 1/n times the number of endofunctions on [n] such that the maximal cardinality of the nonempty preimages equals k. - Alois P. Heinz, May 23 2016
LINKS
FORMULA
T(n,k) = A019575(n,k)/n.
EXAMPLE
: 1;
: 1, 1;
: 2, 6, 1;
: 6, 45, 12, 1;
: 24, 420, 160, 20, 1;
: 120, 4800, 2450, 375, 30, 1;
: 720, 65520, 43050, 7560, 756, 42, 1;
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-j, i-1, k)/j!, j=0..min(k, n))))
end:
T:= (n, k)-> (n-1)!* (b(n$2, k) -b(n$2, k-1)):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jul 29 2014
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k]/j!, {j, 0, Min[k, n]}]]]; T[n_, k_] := (n-1)!*(b[n, n, k]-b[n, n, k-1]); Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 15 2015, after Alois P. Heinz *)
CROSSREFS
Row sums give A000169.
Cf. A019575.
Sequence in context: A191100 A364708 A322944 * A141906 A352125 A136766
KEYWORD
nonn,tabl,easy,nice
AUTHOR
Lee Corbin (lcorbin(AT)tsoft.com)
STATUS
approved