A239761
Number of pairs of functions (f, g) on a set of n elements into itself satisfying f(g(x)) = f(x).
Original entry on oeis.org
1, 1, 10, 159, 3496, 98345, 3373056, 136535455, 6371523712, 336784920849, 19888195110400, 1297716672601151, 92721494240225280, 7199830049013964921, 603715489091812335616, 54366622743565012989375, 5233114241479255004839936, 536180296483497244155041825
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-j, i-1)*binomial(n, j)*j^j, j=0..n)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 17 2014
-
f4[n_] := Sum[n^k Sum[Binomial[n - 1, j]*n^(n - 1 - j)*StirlingS1[j + 1, k] *(-1)^(j + k + 1), {j, 0, n - 1}], {k, 1, n}] (* David Einstein, Oct 31 2016 *)
A239771
Number of pairs of functions (f,g) from a size n set into itself satisfying f(x) = g(g(f(x))).
Original entry on oeis.org
1, 1, 10, 213, 8056, 465945, 37823616, 4075467781, 560230714240, 95369455852497, 19643693349548800, 4805295720474420501, 1374890520609054683136, 454286686896040037996905, 171479277693049020232695808, 73262491601904459123264721125, 35143072854722729593790081499136
Offset: 0
-
g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, k)*Stirling2(n, k)*k!*
add(binomial(n-k, i)*binomial(k, i)*i!*
g(k-i)*n^(n-k-i), i=0..min(k, n-k)), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 18 2014
-
g[n_] := g[n] = If[n < 2, 1, g[n-1] + (n-1)*g[n-2]];
a[n_] := If[n == 0, 1, Sum[Binomial[n, k]*StirlingS2[n, k]*k!*Sum[ Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]} ], {k, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)
A239777
Number of pairs of functions f, g on a size n set into itself satisfying f(g(g(x))) = f(x).
Original entry on oeis.org
1, 1, 12, 249, 7744, 326745, 17773056, 1197261289, 97165842432, 9294416254161, 1030298497753600, 130527793649586201, 18685034341191917568, 2993332161753700720681, 532270629438646194561024, 104316725427708352041239625, 22394627939996943667912769536
Offset: 0
-
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) (l-> add(add(`if`([true$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*)
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 *)
A239770
Number of pairs of functions f, g from a size n set into itself satisfying f(g(f(x))) = g(f(f(x))).
Original entry on oeis.org
1, 1, 10, 213, 7720, 420865, 31879296, 3175850965
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$2);
add(add(`if`([seq(evalb(f[g[f[i]]]=g[f[f[i]]]),
i=1..n)]=[true$n], 1, 0), g=l), f=l)
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
A239774
Number of pairs of functions f, g from a size n set into itself satisfying f(f(g(x))) = f(f(f(x))).
Original entry on oeis.org
1, 1, 10, 285, 14176, 1034145, 105764256, 14367333421
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$2);
add(add(`if`([seq(evalb(f[f[g[i]]]=f[f[f[i]]]),
i=1..n)]=[true$n], 1, 0), g=l), f=l)
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
Showing 1-10 of 12 results.