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.

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

Author

Alois P. Heinz, Dec 19 2021

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Column k=1 gives A350157.

Row sums give A007778.

T(n+1,n) gives A003215 for n>=1.

Cf. A001865, A319298, A322383.

Programs

Maple

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);

Mathematica

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 *)