A293458 Numerator of probability that a permutation of elements of some subset of set {1,2,...,n} is a permutation of elements of some set of the form 1..k, k <= n.

1, 1, 5, 17, 77, 437, 2957, 23117, 204557, 2018957, 21977357, 261478157, 3374988557, 46964134157, 700801318157, 11162196262157, 189005910310157, 3390192763174157, 64212742967590157, 1280663747055910157, 26826134832910630157, 588826498721714470157

Offset: 1

Author

Vladimir Shevelev, Oct 09 2017

Keywords

• nonn
• frac

Comments

The number of all permutations of elements of sets {1..k}, k <= n, is b(n) = Sum_{k=0..n} k! while the number of all permutations of elements of all subsets of set {1,2..n} is c(n) = Sum_{k=0..n} binomial(n,k)!. So the required probability (in a sample space) is b(n)/c(n), n >= 1 (after reduction of the fractions).

Apparently a(n) = A014288(n) for n > 2. - Georg Fischer, Oct 23 2018

Links

• Amiram Eldar, Table of n, a(n) for n = 1..30

Crossrefs

Denominators are in A293459.

Cf. A014288.

Programs

Mathematica

a[n_] := Numerator[Sum[k!, {k, 0, n}]/Sum[Binomial[n, k]!, {k, 0, n}]]; Array[a, 25] (* Amiram Eldar, Sep 21 2019 *)

PARI

a(n) = numerator(sum(k=0, n, k!)/sum(k=0, n, binomial(n,k)!)); \\ Michel Marcus, Oct 12 2017

Extensions

More terms from Peter J. C. Moses, Oct 09 2017