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User: David Einstein

David Einstein has authored 2 sequences.

A275025 Number of pairs of functions (f,g) on [n] such that fg is an idempotent.

1, 1, 14, 411, 21208, 1703145, 195285456, 30113813863, 5985071842688, 1485696848042385, 449588756524844800, 162668114715527356551, 69259775641873646754816, 34243366782512243213286169, 19439795735713938153732810752, 12549399357405863545478828022375

Offset: 0

David Einstein, Nov 12 2016

nonn

			The fourteen pairs of functions on [2] are: ([1,1], [1,1]), ([1,1], [1,2]), ([1,1], [2,1]), ([1,1], [2,2]), ([1,2], [1,1]), ([1,2], [1,2]), ([1,2], [2,2]), ([2,1], [1,1]), ([2,1], [2,1]), ([2,1], [2,2]), ([2,2], [1,1]), ([2,2], [1,2]), ([2,2], [2,1]), ([2,2], [2,2]).

Alois P. Heinz, Table of n, a(n) for n = 0..100
David Einstein, Pseudoinverses on finite sets

Cf. A239768, A000248.

a(n) = Sum_{k = 0..n} ((n!)^2/k!) Sum_{j = 0..n-k} 1/(j!(n-k-j)!) Sum_{l = 0..j} k^(n-k-j+l) n^(n-k-l) stirling2(j,l)/(n-k-l)!.

A277337 Number of pairs of functions (f,g) from a set of n elements into itself that are generalized reflexive inverses of each other.

Original entry on oeis.org

1, 1, 6, 87, 2056, 71145, 3355956, 203899087, 15451934016, 1419181414929, 154796303577700, 19713331210664751, 2891162097251141616, 482733064744447450297, 90871916094948544512516, 19125402877558442317308975, 4467829768503489097383022336, 1151133088512781095709101702177, 325279313240363190497696752254276

Offset: 0

David Einstein, Oct 09 2016

nonn

The number of pairs of functions (f,g) from a set of n elements into itself such that f(g(f(x))) = f(x) and g(f(g(x))) = g(x).

			For n=2 the a(2)=6 solutions are
1: [1,1] [1,1]
2: [1,1] [2,2]
3: [2,2] [1,1]
4: [2,2] [2,2]
5: [1,2] [1,2]
6: [2,1] [2,1]

Alois P. Heinz, Table of n, a(n) for n = 0..272

Cf. A181162, A239749-A239785, A239836-A239841.

Flatten[{1, Table[Sum[n!*Binomial[n, k]*k^(2*(n-k))/(n-k)!, {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 21 2016 *)

a(n) = sum(k = 1, n, n! / (n - k)! * binomial(n, k) * k^(2 * (n - k) ) ); \\ Joerg Arndt, Oct 10 2016

a(n) = Sum_{k=0..n} ((n! / (n - k)!) * C(n, k) * k^(2 * (n - k))).

a(0)=1 prepended by Alois P. Heinz, Oct 20 2016

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User: David Einstein

A275025 Number of pairs of functions (f,g) on [n] such that fg is an idempotent.

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A277337 Number of pairs of functions (f,g) from a set of n elements into itself that are generalized reflexive inverses of each other.

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