A332854 - OEIS

%I #8 Mar 13 2021 10:04:34

%S 1,16,197,2311,27568,343909,4541329,63719723,949770615,15010838233,

%T 250997692441,4430433962701,82376202579421,1610014961936672,

%U 33010385435710028,708642421376230354,15899009565671538930,372166745683645768206,9074749796104015627262

%N Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by decreasing lengths.

%H Alois P. Heinz, <a href="/A332854/b332854.txt">Table of n, a(n) for n = 5..450</a>

%H Andrew V. Sills, <a href="https://arxiv.org/abs/1912.05306">Integer Partitions Probability Distributions</a>, arXiv:1912.05306 [math.CO], 2019.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

%p b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

%p       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*

%p         b(n-i*j, min(n-i*j, i-1), max(0, t-j))/j!*

%p         combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2, 5)[2]:

%p seq(a(n), n=5..23);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i<1, {0, 0},

%t      Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i},

%t      {0, 0}]][(i-1)!^j*b[n - i*j, Min[n - i*j, i-1], Max[0, t-j]]/j!*

%t      multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];

%t a[n_] := b[n, n, 5][[2]];

%t Table[a[n], {n, 5, 23}] (* _Jean-François Alcover_, Mar 13 2021, after _Alois P. Heinz_ *)

%Y Column k=5 of A322384.

%K nonn

%O 5,2

%A _Alois P. Heinz_, Feb 26 2020