A209327 -id:A209327 - cp's OEIS frontend

A209324 Triangular array read by rows: T(n,k) is the number of endofunctions f:{1,2,...,n}-> {1,2,...,n} whose largest component has exactly k nodes; n>=1, 1<=k<=n.

1, 1, 3, 1, 9, 17, 1, 45, 68, 142, 1, 165, 680, 710, 1569, 1, 855, 6290, 8520, 9414, 21576, 1, 3843, 47600, 134190, 131796, 151032, 355081, 1, 21819, 481712, 1838900, 2372328, 2416512, 2840648, 6805296, 1, 114075, 5025608, 21488292, 50609664, 48934368, 51131664, 61247664, 148869153

Offset: 1

Geoffrey Critzer, Jan 19 2013

Here component means weakly connected component in the functional digraph of f.
Row sums are n^n.
T(n,n) = A001865.
For the statistic "length of the smallest component", see A347999.

			Triangle T(n,k) begins:
  1;
  1,     3;
  1,     9,     17;
  1,    45,     68,     142;
  1,   165,    680,     710,    1569;
  1,   855,   6290,    8520,    9414,   21576;
  1,  3843,  47600,  134190,  131796,  151032,  355081;
  1, 21819, 481712, 1838900, 2372328, 2416512, 2840648, 6805296;
  ...

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, Chapter 8.

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Main diagonal gives A001865.
Row sums give A000312.
Cf. A209327, A347999.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*
      b(n-i, max(m, i))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):
seq(T(n), n=1..12);  # Alois P. Heinz, Dec 16 2021

nn=8;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];c=Log[1/(1-t)];b=Drop[Range[0,nn]!CoefficientList[Series[c,{x,0,nn}],x],1];f[list_]:=Select[list,#>0&];Map[f,Drop[Transpose[Table[Range[0,nn]!CoefficientList[Series[ Exp[Sum[b[[i]]x^i/i!,{i,1,n+1}]]-Exp[Sum[b[[i]]x^i/i!,{i,1,n}]],{x,0,nn}],x],{n,0,nn-1}]],1]]//Grid
(* Second program: *)
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*b[n - i, Max[m, i]]* Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, 0]}, Table[Coefficient[p, x, i], {i, 1, n}]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 30 2021, after Alois P. Heinz *)

E.g.f. for column k: exp( Sum_{n=1..k} A001865(n) x^n/n!) - exp( Sum_{n=1..k-1} A001865(n) x^n/n!).

Sum_{k=1..n} k * T(n,k) = A209327(n). - Alois P. Heinz, Dec 16 2021

A350157 Total number of nodes in the smallest connected component summed over all endofunctions on [n].

Original entry on oeis.org

0, 1, 7, 61, 709, 9911, 167111, 3237921, 71850913, 1780353439, 49100614399, 1482061739423, 48873720208853, 1740252983702871, 66793644836081827, 2740470162691675711, 120029057782404141841, 5575505641199441262767, 274412698693082818767335, 14236421024010426118259883

Offset: 0

Alois P. Heinz, Dec 17 2021

nonn

			a(2) = 7 = 2 + 2 + 1 + 2: 11, 22, 12, 21.

Alois P. Heinz, Table of n, a(n) for n = 0..385

Column k=1 of A350202.

Cf. A000312, A001865, A007778, A209327, A347999.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(
      b(n-i, min(m, i))*g(i)*binomial(n-1, i-1), i=1..n))
    end:
a:= n-> (p-> add(coeff(p, x, i)*i, i=0..n))(b(n,n)):
seq(a(n), n=0..23);

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[
     b[n - i, Min[m, i]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
a[n_] := Function[p, Sum[Coefficient[p, x, i]*i, {i, 0, n}]][b[n, n]];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A347999(n,k).

cp's OEIS Frontend

A209324 Triangular array read by rows: T(n,k) is the number of endofunctions f:{1,2,...,n}-> {1,2,...,n} whose largest component has exactly k nodes; n>=1, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula