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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.
2

%I #28 Feb 11 2020 02:04:13

%S 1,1,5,17,77,437,2957,23117,204557,2018957,21977357,261478157,

%T 3374988557,46964134157,700801318157,11162196262157,189005910310157,

%U 3390192763174157,64212742967590157,1280663747055910157,26826134832910630157,588826498721714470157

%N 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.

%C 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).

%C Apparently a(n) = A014288(n) for n > 2. - _Georg Fischer_, Oct 23 2018

%H Amiram Eldar, <a href="/A293458/b293458.txt">Table of n, a(n) for n = 1..30</a>

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

%o (PARI) a(n) = numerator(sum(k=0, n, k!)/sum(k=0, n, binomial(n,k)!)); \\ _Michel Marcus_, Oct 12 2017

%Y Denominators are in A293459.

%Y Cf. A014288.

%K nonn,frac

%O 1,3

%A _Vladimir Shevelev_, Oct 09 2017

%E More terms from _Peter J. C. Moses_, Oct 09 2017