A332912
Number of entries in the eighth cycles of all permutations of [n] when cycles are ordered by increasing lengths.
Original entry on oeis.org
1, 73, 2026, 51866, 1205832, 26820444, 590148378, 13100356128, 295439599773, 6807481609901, 160887057723746, 3910224803769926, 97864814302511668, 2524228366327596948, 67113790045849483056, 1839604074821982820944, 51976021537896649002555
Offset: 8
-
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
(n, i$j, n-i*j)), j=0..n/i)))
end:
a:= n-> b(n, 1, 8)[2]:
seq(a(n), n=8..24);
A332913
Number of entries in the ninth cycles of all permutations of [n] when cycles are ordered by increasing lengths.
Original entry on oeis.org
1, 91, 3026, 90597, 2439516, 61989747, 1538707458, 38139479553, 953072924061, 24163160247008, 624376555920980, 16498591390522978, 446749936369626468, 12412653806246040786, 354123558880901036688, 10378353585187092249138, 312512449355524805863371
Offset: 9
-
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
(n, i$j, n-i*j)), j=0..n/i)))
end:
a:= n-> b(n, 1, 9)[2]:
seq(a(n), n=9..25);
A332914
Number of entries in the tenth cycles of all permutations of [n] when cycles are ordered by increasing lengths.
Original entry on oeis.org
1, 111, 4357, 150801, 4654469, 134049203, 3729595473, 102656565841, 2828539643998, 78576462334232, 2212108223706338, 63356175294304198, 1851016172004724086, 55265191691761995338, 1688128671403459821842, 52796230233728968111746, 1691419149055825399281421
Offset: 10
-
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
(n, i$j, n-i*j)), j=0..n/i)))
end:
a:= n-> b(n, 1, 10)[2]:
seq(a(n), n=10..26);
A350202
Number T(n,k) of nodes in the k-th connected component of all endofunctions on [n] when components are ordered by increasing size; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
Original entry on oeis.org
1, 7, 1, 61, 19, 1, 709, 277, 37, 1, 9911, 4841, 811, 61, 1, 167111, 91151, 19706, 1876, 91, 1, 3237921, 1976570, 486214, 60229, 3739, 127, 1, 71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1, 1780353439, 1257567127, 380291461, 62248939, 5971291, 340729, 11197, 217, 1
Offset: 1
Triangle T(n,k) begins:
1;
7, 1;
61, 19, 1;
709, 277, 37, 1;
9911, 4841, 811, 61, 1;
167111, 91151, 19706, 1876, 91, 1;
3237921, 1976570, 486214, 60229, 3739, 127, 1;
71850913, 47203241, 13110749, 1892997, 152937, 6721, 169, 1;
...
-
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(g(i)^j*
b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
(n, i$j, n-i*j)), j=0..n/i)))
end:
T:= (n, k)-> b(n, 1, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..10);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][g[i]^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
T[n_, k_] := b[n, 1, k][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)