A239750
Number of ordered pairs of endofunctions (f,g) on a set of n elements satisfying g(f(x)) = f(f(f(x))).
Original entry on oeis.org
1, 1, 6, 87, 2200, 84245, 4492656, 315937195, 28186856832, 3099006365769, 410478164588800, 64323095036300111, 11748771067445148672, 2470422069374379054493, 591735532838657160296448, 160004357420756572368889875, 48458574881000820765562863616
Offset: 0
-
a:= n-> add(binomial(n, k)*k^n*(n-1)^(n-k), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 23 2014
-
a[n_] := If[n<2, 1, Sum[Binomial[n, k]*k^n*(n-1)^(n-k), {k, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)
A239768
Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(x) = f(g(f(x))).
Original entry on oeis.org
1, 1, 10, 195, 6808, 362745, 26848656, 2621263519, 324981308800, 49669569764433, 9146879704748800, 1993011341241988551, 506190915590699695104, 148000190814308473203433, 49289886405448749446514688, 18529196186934893511062427375, 7800708229072237749055062900736, 3652486190893312491910941333813537
Offset: 0
-
A[n_] := If[n == 0, 1, Sum[(n!/(n - k)!) Binomial[n, k] (n k)^(n - k), {k, 1, n}]]
Table[A[n],{n,10}] (* David Einstein, Oct 10 2016 *)
-
a(n) = sum(k= 0, n, (n!/(n-k)!)*binomial(n,k)*(n*k)^(n-k)); \\ Michel Marcus, Oct 11 2016; corrected Jun 13 2022
A239777
Number of pairs of functions f, g on a size n set into itself satisfying f(g(g(x))) = f(x).
Original entry on oeis.org
1, 1, 12, 249, 7744, 326745, 17773056, 1197261289, 97165842432, 9294416254161, 1030298497753600, 130527793649586201, 18685034341191917568, 2993332161753700720681, 532270629438646194561024, 104316725427708352041239625, 22394627939996943667912769536
Offset: 0
-
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
f[g[g[i]]]=f[i]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!
*x^((2-irem(i, 2))*j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> add((p-> add(n^i*binomial(n-1, k-1)*n^(n-k)*
coeff(p, x, i), i=0..degree(p)))(b(k$2)), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 06 2014
-
c[n_] := c[n] =
Sum[(n - 1)! n^(n - k)/(n - k)! t^(1 + Mod[k + 1, 2]), {k, 1, n}]
d[0] = 1
d[n_] := d[n] = Sum[Binomial[n - 1, k]*d[k]*c[n - k], {k, 0, n - 1}]
a[n_] := d[n] /. t -> n
Table[a[n], {n, 1, 10}] (* David Einstein, Nov 02 2016*)
A254529
a(n) = n! * (number of mapping patterns on n).
Original entry on oeis.org
1, 1, 6, 42, 456, 5640, 93600, 1728720, 38344320, 948931200, 26555558400, 817935148800, 27735629644800, 1020596255078400, 40642432179148800, 1737890081351424000, 79498734605402112000, 3871319396080840704000, 200017645344178421760000, 10925549584125028909056000
Offset: 0
A254570
The number of unordered 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) where f and g are distinct.
Original entry on oeis.org
0, 3, 57, 1284, 34220, 1098720, 41579328, 1832244288, 92830006368, 5353120671120, 348383876993900, 25409389391925264, 2064511110000765192, 185885772163424273304, 18458953746901624026000, 2012589235930543617012480, 239897773975844015012351360, 31132547318002718989156350240, 4380969784826872849927354999092, 665896601825393760478978112600400
Offset: 1
The a(2) = 3 pairs of maps [2] -> [2] are:
01: [ 1 1 ] [ 1 2 ]
02: [ 1 2 ] [ 2 1 ]
03: [ 1 2 ] [ 2 2 ]
A239772
Number of pairs of functions f, g from a size n set into itself satisfying f(f(x)) = f(g(f(x))).
Original entry on oeis.org
1, 1, 10, 231, 9688, 603445, 52284816, 5951141035, 856275088768, 151330313546361, 32121886627244800, 8043522214887251191, 2341436450503523834880, 782684599861773582454741, 297337340445195054893615104, 127232791559907423447708979875, 60852096942278280426353043275776, 32309821732254010064727052008198385
Offset: 0
-
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) local l; l:= s(n$2);
add(add(`if`([seq(evalb(f[f[i]]=f[g[f[i]]]),
i=1..n)]=[true$n], 1, 0), g=l), f=l)
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
-
def a239772(n):
L. = LaurentPolynomialRing(QQ)
R. = PowerSeriesRing(L, default_prec=n+1)
h = 1 - sum((y*(1+i*z))^i*n^(i-1)/factorial(i) for i in (1..n))//z
return h.inverse()[n][0] * factorial(n) # Max Alekseyev, Jan 10 2025
A239775
Number of pairs of functions f, g from a size n set into itself satisfying f(f(g(x))) = f(f(x)).
Original entry on oeis.org
1, 1, 10, 297, 13264, 851325, 74078496, 8325102331, 1169885964640, 200545429514697, 41101718746949920
Offset: 0
-
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
f[f[g[i]]]=f[f[i]]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
-
# from Max Alekseyev, Dec 19 2024
from collections import Counter
def a239775(n): return sum( prod(k^k for k in Counter(f[t] for t in f).values()) for f in Tuples(range(n),n) )
A239778
Number of pairs of functions f, g from a size n set into itself satisfying f(f(f(x))) = f(g(g(x))).
Original entry on oeis.org
1, 1, 12, 255, 8968, 452485, 31456656, 2899786855, 343386848064
Offset: 0
-
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
f[f[f[i]]]=f[g[g[i]]]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
A239779
Number of pairs of functions f, g, from a size n set into itself satisfying f(g(g(x))) = g(g(f(x))).
Original entry on oeis.org
1, 1, 12, 267, 9088, 425465, 27039096, 2261637637
Offset: 0
-
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
f[g[g[i]]]=g[g[f[i]]]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
A239781
Number of pairs of functions f, g from a size n set into itself satisfying f(g(x)) = f(f(g(x))).
Original entry on oeis.org
1, 1, 12, 321, 15280, 1127745, 118507536, 16731979033, 3044595017472, 692050790547297, 191796657547052800, 63563842088104098081, 24793529117087476242432, 11232023076988690608825505, 5843573099019743656060348416, 3457799186387568447755745563625
Offset: 0
-
a:= n-> add(binomial(n, k)*add(binomial(n-k, i)*k^i*
(n-k-1)^(n-k-i)*(k+i)^n, i=0..n-k), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 17 2014
-
Unprotect[Power]; 0^0 = 1; a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n-k, i]*k^i*(n-k-1)^(n-k-i)*(k+i)^n, {i, 0, n-k}], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)
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