A201685 Triangular array read by rows. T(n,k) is the number of connected endofunctions on {1,2,...,n} that have exactly k nodes in the unique cycle of its digraph representation.

1, 2, 1, 9, 6, 2, 64, 48, 24, 6, 625, 500, 300, 120, 24, 7776, 6480, 4320, 2160, 720, 120, 117649, 100842, 72030, 41160, 17640, 5040, 720, 2097152, 1835008, 1376256, 860160, 430080, 161280, 40320, 5040, 43046721, 38263752, 29760696, 19840464, 11022480, 4898880, 1632960, 362880, 40320

Offset: 1

Author

Geoffrey Critzer, Dec 03 2011

Keywords

• nonn
• tabl

Comments

Column k=1: A000169,

Column k=2: A053506,

Column k=3: A065513.

Row sums: A001865.

T(n,n) = (n-1)!, T(n,n-1) = n!.

Sum_{k=1..n} T(n,k)*k = n^n. - Geoffrey Critzer, May 13 2013

From the asymptotic given by N-E. Fahssi in A001865, we see the expected size of the cycle grows as (2*n/Pi)^(1/2). - Geoffrey Critzer, May 13 2013

Central terms: A277168. - Paul D. Hanna, Oct 01 2016

Examples

			Triangle begins as:
     1;
     2,    1;
     9,    6,    2;
    64,   48,   24,    6;
   625,  500,  300,  120,  24;
  7776, 6480, 4320, 2160, 720, 120;
		

Links

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

Crossrefs

Cf. A000169, A001865, A053506, A065513, A277168.

Programs

GAP

Flat(List([1..12], n-> List([1..n], k-> Binomial(n-1,k-1)*n^(n-k)*Factorial(k-1) ))); # G. C. Greubel, Jan 08 2020

Magma

[Binomial(n-1,k-1)*n^(n-k)*Factorial(k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 08 2020

Maple

T:= (n, k)-> binomial(n-1, k-1)*n^(n-k)*(k-1)!:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 14 2013

Mathematica

f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Map[f, Drop[Range[0, 10]! CoefficientList[Series[Log[1/(1 - y t)], {x, 0, 10}], {x, y}], 1]] // Grid

PARI

T(n,k) = binomial(n-1,k-1)*n^(n-k)*(k-1)!; \\ G. C. Greubel, Jan 08 2020

Sage

[[binomial(n-1,k-1)*n^(n-k)*factorial(k-1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jan 08 2020

Formula

E.g.f.: log(1/(1-y*A(x))) where A(x) is the e.g.f. for A000169.

T(n,k) = binomial(n-1,k-1)*n^(n-k)*(k-1)!. - Geoffrey Critzer, May 13 2013