A285439 Sum T(n,k) of the entries in the k-th cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
1, 4, 2, 21, 12, 3, 132, 76, 28, 4, 960, 545, 235, 55, 5, 7920, 4422, 2064, 612, 96, 6, 73080, 40194, 19607, 6692, 1386, 154, 7, 745920, 405072, 202792, 75944, 18736, 2816, 232, 8, 8346240, 4484808, 2280834, 911637, 254061, 46422, 5256, 333, 9
Offset: 1
Examples
T(3,1) = 21 because the sum of the entries in the first cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+4+1+1 = 21.
Triangle T(n,k) begins:
1;
4, 2;
21, 12, 3;
132, 76, 28, 4;
960, 545, 235, 55, 5;
7920, 4422, 2064, 612, 96, 6;
73080, 40194, 19607, 6692, 1386, 154, 7;
745920, 405072, 202792, 75944, 18736, 2816, 232, 8;
...
Links
- Alois P. Heinz, Rows n = 1..23, flattened
- Wikipedia, Permutation
Crossrefs
Programs
-
Maple
T:= proc(h) option remember; local b; b:= proc(n, l) option remember; `if`(n=0, [mul((i-1)!, i=l), 0], (p-> p+[0, (h-n+1)*p[1]*x^(nops(l)+1)])(b(n-1, [l[], 1]))+ add((p-> p+[0, (h-n+1)*p[1]*x^j])( b(n-1, subsop(j=l[j]+1, l))), j=1..nops(l))) end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2]) end: seq(T(n), n=1..10); -
Mathematica
T[h_] := T[h] = Module[{b}, b[n_, l_] := b[n, l] = If[n == 0, {Product[(i - 1)!, {i, l}], 0}, # + {0, (h - n + 1)*#[[1]]*x^(Length[l] + 1)}&[b[n - 1, Append[l, 1]]] + Sum[# + {0, (h-n+1)*#[[1]]*x^j}&[b[n - 1, ReplacePart[ l, j -> l[[j]] + 1]]], {j, 1, Length[l]}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, {}][[2]]]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)
Formula
Sum_{k=1..n} k * T(n,k) = n^2 * n! = A002775(n).
Comments