A131104 -id:A131104 - cp's OEIS frontend

A131106 Rectangular array read by antidiagonals: k objects are each put into one of n boxes, independently with equal probability. a(n, k) is the expected number of boxes with exactly one object (n, k >= 1). Sequence gives the numerators.

1, 1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 3, 4, 1, 0, 1, 8, 27, 32, 5, 0, 1, 5, 48, 27, 80, 3, 0, 1, 12, 25, 256, 405, 64, 7, 0, 1, 7, 108, 125, 256, 729, 448, 1, 0, 1, 16, 147, 864, 3125, 6144, 5103, 1024, 9, 0, 1, 9, 64, 343, 6480, 3125, 28672, 2187, 256, 5, 0, 1, 20, 243, 2048, 12005

Offset: 1

David Wasserman, Jun 15 2007

Problem suggested by Brandon Zeidler. To motivate this sequence, suppose that when objects are placed in the same box, they mix and the information they contain is lost. The sequence tells us how much information we can expect to recover.

			Array begins:
  1 0 0 0 0 0 ...
  1 1 3/4 1/2 5/16 3/16 ...
  1 4/3 4/3 32/27 80/81 64/81 ...
  ...

Cf. A131107 gives the denominators. A131103, A131104 and A131105 give f(n, k, 0), f(n, k, 1) and f(n, k, 2).

a(n, k) = k*(1 - 1/n)^(k - 1). Let f(n, k, i) be the number of assignments such that exactly i boxes have exactly one object. For i > n, f(n, k, i) = 0. For i = k <= n, f(n, k, i) = n!/(n-k)!. Otherwise, f(n, k, i) = Sum_{j = 1..min(floor((k-i)/2), n-i)} A008299(k-i, j)*n!*binomial(k, i)/(n-i-j)!. Then a(n, k) = Sum_{i=1..min(n, k)} i*f(n, k, i)/n^k.

A131103 Rectangular array read by antidiagonals: a(n, k) is the number of ways to put k labeled objects into n labeled boxes so that there are no boxes with exactly one object (n, k >= 1).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 8, 1, 0, 5, 4, 21, 22, 1, 0, 6, 5, 40, 63, 52, 1, 0, 7, 6, 65, 124, 243, 114, 1, 0, 8, 7, 96, 205, 664, 969, 240, 1, 0, 9, 8, 133, 306, 1405, 3196, 3657, 494, 1, 0, 10, 9, 176, 427, 2556, 7425, 15712, 12987, 1004, 1, 0, 11, 10, 225, 568, 4207

Offset: 1

David Wasserman, Jun 14 2007, Jun 15 2007

Problem suggested by Brandon Zeidler. Columns four and five are A000567 and A051874. Second row is A130102.

			Array begins:
0 1 1 1 1 1 1
0 2 2 8 22 52 114
0 3 3 21 63 243 969

Cf. A131104, A131105, A131106, A131107.

a(n, k) = sum_{j=1..min(floor(k/2), n)} A008299(k, j)*n!/(n-j)!.

A131105 Rectangular array read by antidiagonals: a(n, k) is the number of ways to put k labeled objects into n labeled boxes so that there are exactly two boxes with exactly one object (n, k >= 2).

Original entry on oeis.org

2, 6, 0, 12, 0, 0, 20, 0, 36, 0, 30, 0, 144, 60, 0, 42, 0, 360, 240, 90, 0, 56, 0, 720, 600, 1440, 126, 0, 72, 0, 1260, 1200, 6300, 5544, 168, 0, 90, 0, 2016, 2100, 18000, 26460, 17472, 216, 0, 110, 0, 3024, 3360, 40950, 78120, 136080, 49248, 270, 0, 132, 0, 4320

Offset: 2

David Wasserman, Jun 15 2007

Problem suggested by Brandon Zeidler. Columns 2, 4 and 5 are A002378, 36*A000292 and 60*A000292.

			Array begins:
2 0 0 0 0 0
6 0 36 60 90 126
12 0 144 240 1440 5544

Cf. A131103, A131104, A131106, A131107.

a(n, 2) = n^2-n. For k > 2, a(n, k) = sum_{j=1..min(floor(k/2)-1, n-2)} A008299(k-2, j)*n!*(k^2-k)/(2*(n-j-2)!).

cp's OEIS Frontend

A131106 Rectangular array read by antidiagonals: k objects are each put into one of n boxes, independently with equal probability. a(n, k) is the expected number of boxes with exactly one object (n, k >= 1). Sequence gives the numerators.

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A131103 Rectangular array read by antidiagonals: a(n, k) is the number of ways to put k labeled objects into n labeled boxes so that there are no boxes with exactly one object (n, k >= 1).

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Formula

A131105 Rectangular array read by antidiagonals: a(n, k) is the number of ways to put k labeled objects into n labeled boxes so that there are exactly two boxes with exactly one object (n, k >= 2).

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Author

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Comments

Examples

Crossrefs

Formula