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

1,2

Links

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

Example

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;
  ...

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 *)

Crossrefs

Column k=1 gives A350157.

Row sums give A007778.

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

Cf. A001865, A319298, A322383.

Sequence in context: A144450 A051339 A134141 * A237111 A281620 A321187

Adjacent sequences: A350199 A350200 A350201 * A350203 A350204 A350205

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 19 2021

Status

approved