A246049
Number T(n,k) of endofunctions on [n] where the smallest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 19, 6, 2, 0, 175, 51, 24, 6, 0, 2101, 580, 300, 120, 24, 0, 31031, 8265, 4360, 2160, 720, 120, 0, 543607, 141246, 74130, 41160, 17640, 5040, 720, 0, 11012415, 2810437, 1456224, 861420, 430080, 161280, 40320, 5040
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 19, 6, 2;
0, 175, 51, 24, 6;
0, 2101, 580, 300, 120, 24;
0, 31031, 8265, 4360, 2160, 720, 120;
0, 543607, 141246, 74130, 41160, 17640, 5040, 720;
...
Columns k=0-10 give:
A000007,
A045531,
A246189,
A246190,
A246191,
A246192,
A246193,
A246194,
A246195,
A246196,
A246197.
Main diagonal gives
A000142(n-1) for n>0.
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i+1), j=0..n/i)))
end:
A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), A(n, k) -A(n, k+1)):
seq(seq(T(n, k), k=0..n), n=0..12);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0,
Sum[(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!*
b[n-i*j, i+1], {j, 0, n/i}]]];
A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, k], {j, 0, n}];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], A[n, k] - A[n, k+1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)
A243098
Number T(n,k) of endofunctions on [n] with all cycles of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 16, 6, 2, 0, 125, 51, 24, 6, 0, 1296, 560, 300, 120, 24, 0, 16807, 7575, 4360, 2160, 720, 120, 0, 262144, 122052, 73710, 41160, 17640, 5040, 720, 0, 4782969, 2285353, 1430016, 861420, 430080, 161280, 40320, 5040
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 16, 6, 2;
0, 125, 51, 24, 6;
0, 1296, 560, 300, 120, 24;
0, 16807, 7575, 4360, 2160, 720, 120;
0, 262144, 122052, 73710, 41160, 17640, 5040, 720;
...
Main diagonal gives
A000142(n-1) for n>0.
-
with(combinat):
T:= (n, k)-> `if`(k*n=0, `if`(k+n=0, 1, 0),
add(binomial(n-1, j*k-1)*n^(n-j*k)*(k-1)!^j*
multinomial(j*k, k$j, 0)/j!, j=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..10);
-
multinomial[n_, k_] := n!/Times @@ (k!); T[n_, k_] := If[k*n==0, If[k+n == 0, 1, 0], Sum[Binomial[n-1, j*k-1]*n^(n-j*k)*(k-1)!^j*multinomial[j*k, Append[Array[k&, j], 0]]/j!, {j, 0, n/k}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)
A241982
Number of endofunctions on [2n] where the largest cycle length equals n.
Original entry on oeis.org
1, 3, 93, 8600, 1719060, 604727424, 331079253120, 260480095349760, 278592031202284800, 388855261570122547200, 686533182382689959116800, 1495779844806108697677004800, 3942052104672989614027181260800, 12360865524060039746012601384960000
Offset: 0
a(1) = 3: (1,1), (1,2), (2,2).
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1), j=0..n/i)))
end:
A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
a:= n-> `if`(n=0, 1, A(2*n, n) -A(2*n, n-1)):
seq(a(n), n=0..15);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Which[n==0, 1, i<1, 0, True, Sum[(i-1)!^j* multinomial[n, Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1], {j, 0, n/i} ] ];
A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, Min[j, k]], {j, 0, n}];
a[n_] := If[n == 0, 1, A[2n, n] - A[2n, n-1]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 01 2017, translated from Maple *)
Showing 1-3 of 3 results.
Comments