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 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). 35
 1, 1, 10, 141, 2824, 71565, 2244096, 83982199, 3681265792, 186047433225, 10716241342240, 697053065658411, 50827694884298784, 4129325095108122637, 371782656333674104624, 36918345387693628911375, 4025196918605160943576576, 479796375191949916361466897 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 LINKS Martin Fuller, Table of n, a(n) for n = 0..20 Joerg Arndt, the a(3) = 141 pairs of maps  ->  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. EXAMPLE The a(2) = 10 pairs of maps  ->  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 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 *) CROSSREFS A053529 is a similar count for permutations. A254529 is for permutations commuting with functions. Cf. A000312, A001372, A239749-A239785, A239836-A239841. Sequence in context: A277310 A343689 A277372 * A245988 A184710 A263055 Adjacent sequences: A181159 A181160 A181161 * A181163 A181164 A181165 KEYWORD hard,nonn,nice AUTHOR Jeffrey Norden, Oct 07 2010 EXTENSIONS a(11)-a(20) from Martin Fuller, Feb 01 2015 STATUS approved

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Last modified March 30 00:49 EDT 2023. Contains 361599 sequences. (Running on oeis4.)