A053529
a(n) = n! * number of partitions of n.
Original entry on oeis.org
1, 1, 4, 18, 120, 840, 7920, 75600, 887040, 10886400, 152409600, 2235340800, 36883123200, 628929100800, 11769069312000, 230150688768000, 4833164464128000, 105639166144512000, 2464913876705280000, 59606099200327680000, 1525429559126753280000, 40464026199993876480000
Offset: 0
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.12, solution.
- T. D. Noe, Table of n, a(n) for n = 0..200
- M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, PU.M.A, Volume 24 (2013), Issue No. 1.
- M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, Research Gate, 2015.
- R. P. Stanley, Pairs with equal squares, Problem 10654, Amer. Math. Monthly, 107 (April 2000), solution p. 368.
- Wikipedia, Young tableau
-
a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // Vincenzo Librandi, Jan 17 2019
-
seq(count(Permutation(n))*count(Partition(n)),n=1..20); # Zerinvary Lajos, Oct 16 2006
with(combinat): A053529 := proc(n): n! * numbpart(n) end: seq(A053529(n), n=0..20); # Johannes W. Meijer, Jul 28 2016
-
Table[PartitionsP[n] n!, {n, 0, 20}] (* T. D. Noe, Jun 19 2012 *)
-
N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, x^k/(1-x^k)/k)))) \\ Joerg Arndt, Apr 16 2010
-
N=66; x='x+O('x^N); Vec(serlaplace(sum(n=0, N, x^n/prod(k=1,n,1-x^k)))) \\ Joerg Arndt, Jan 29 2011
-
a(n) = n!*numbpart(n); \\ Michel Marcus, Jul 28 2016
-
from math import factorial
from sympy import npartitions
def A053529(n): return factorial(n)*npartitions(n) # Chai Wah Wu, Jul 10 2023
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).
Original entry on oeis.org
1, 1, 10, 141, 2824, 71565, 2244096, 83982199, 3681265792, 186047433225, 10716241342240, 697053065658411, 50827694884298784, 4129325095108122637, 371782656333674104624, 36918345387693628911375, 4025196918605160943576576, 479796375191949916361466897
Offset: 0
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
- 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.
A053529 is a similar count for permutations.
A254529 is for permutations commuting with functions.
-
(* 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 *)
A239749
Number of ordered pairs of functions f,g on a set of n elements into itself satisfying f(f(x)) = g(f(g(x))).
Original entry on oeis.org
1, 1, 6, 87, 2056, 69605, 3201696, 190933435
Offset: 0
a(0) = a(1) = 1 since there is only one endofunction for n=0 or 1 and the equation is satisfied trivially. For n=2, each endofunction f on {1,2} is represented by [f(1),f(2)]. The list of a(2) = 6 pairs (f,g) which satisfy the equation is ([1,1], [1,1]), ([1,1], [1,2]), ([1,2], [1,2]), ([1,2], [2,1]), ([2,2], [1,2]), ([2,2], [2,2]). - _Michael Somos_, Mar 26 2014
A255525
1/n! times the number of ordered pairs of permutation functions f,g on n elements where f(g(g(x))) = g(f(f(x))).
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 4, 4, 6, 9, 13, 17, 27, 31, 42, 57, 83, 104
Offset: 0
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
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]
-
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
Showing 1-5 of 5 results.
Comments