A181162 Number of commuting functions: the number of ordered pairs (f,g) of functions from {1..n} to itself such that fg=gf (i.e., f(g(i))=g(f(i)) for all i).

1, 1, 10, 141, 2824, 71565, 2244096, 83982199, 3681265792, 186047433225, 10716241342240, 697053065658411, 50827694884298784, 4129325095108122637, 371782656333674104624, 36918345387693628911375, 4025196918605160943576576, 479796375191949916361466897

Offset: 0

Author

Jeffrey Norden, Oct 07 2010

Keywords

• hard
• nonn
• nice

Comments

Also, the total number of endomorphisms of all directed graphs on n labeled vertices with outdegree of each vertex equal 1. - Max Alekseyev, Jan 09 2015

Seems to be relatively hard to compute for large n. (a(n)-n^n)/2 is always an integer, since it gives the number of unordered pairs of distinct commuting functions.

a(n) is divisible by n as proved by Holloway and Shattuck (2015).

From Joerg Arndt, Jul 21 2014: (Start)

Multiply fg=gf from the right by f to obtain fgf=gff, and use f(gf)=f(fg)=ffg to see ffg=gff; iterate to see f^k g = g f^k for all k>=1; by symmetry g^k f = f g^k holds as well.

More generally, if X and Y are words of length w over the alphabet {f,g}, then X = Y (as functional composition) whenever both words contain j symbols f and k symbols g (and j+k=w). (End)

Functions with the same mapping pattern have the same number of commuting functions, so there is no need to check every pair. - Martin Fuller, Feb 01 2015

Examples

			The a(2) = 10 pairs of maps [2] -> [2] are:
01:  [ 1 1 ]  [ 1 1 ]
02:  [ 1 1 ]  [ 1 2 ]
03:  [ 1 2 ]  [ 1 1 ]
04:  [ 1 2 ]  [ 1 2 ]
05:  [ 1 2 ]  [ 2 1 ]
06:  [ 1 2 ]  [ 2 2 ]
07:  [ 2 1 ]  [ 1 2 ]
08:  [ 2 1 ]  [ 2 1 ]
09:  [ 2 2 ]  [ 1 2 ]
10:  [ 2 2 ]  [ 2 2 ]
- _Joerg Arndt_, Jul 22 2014
		

Links

• Martin Fuller, Table of n, a(n) for n = 0..20
• Joerg Arndt, the a(3) = 141 pairs of maps [3] -> [3]
• Stephen M. Buckley, Minimal order semigroups with specified commuting probability, 04-03-2013. - _W. Edwin Clark_, Jul 21 2014
• Martin Fuller, a(6) from the A001372(6)=130 mapping patterns
• M. Holloway and M. Shattuck, Commuting pairs of functions on a finite set, 2015.
• Math Overflow, What is the probability two random maps on n symbols commute?, 2013. - _W. Edwin Clark_, Jul 21 2014
• Math Overflow, Counting and understanding commuting functions, 2010.

Crossrefs

A053529 is a similar count for permutations. A254529 is for permutations commuting with functions.

Cf. A000312, A001372, A239749-A239785, A239836-A239841.

Programs

Mathematica

(* This brute force code allows to get a few terms *)
a[n_] := a[n] = If[n == 0, 1, Module[{f, g, T}, T = Tuples[Range[n], n]; Table[f = T[[j, #]]&; g = T[[k, #]] &; Table[True, {n}] == Table[f[g[i]] == g[f[i]], {i, n}], {j, n^n}, {k, n^n}] // Flatten // Count[#, True]&]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 5}] (* Jean-François Alcover, Sep 24 2022 *)

Extensions

a(11)-a(20) from Martin Fuller, Feb 01 2015