A341093 Triangular array read by rows. T(n,k) is the number of partial functions on [n] with index k, n=0 implies k=1, otherwise n >= 1, 1 <= k <= n.

1, 2, 7, 2, 37, 21, 6, 261, 232, 108, 24, 2301, 2935, 1760, 660, 120, 24343, 42396, 30630, 14880, 4680, 720, 300455, 692055, 586572, 335790, 139440, 37800, 5040, 4238153, 12631200, 12387592, 8008896, 3959760, 1438080, 342720, 40320

Offset: 0

Author

Geoffrey Critzer, Feb 13 2022

Keywords

• nonn
• tabf

Comments

For every partial function f, there are smallest positive integers k,m such that f^k = f^(k+m). The integer k is the index of f.

Examples

			Array begins
      1;
      2;
      7,     2;
     37,    21,     6;
    261,   232,   108,    24;
   2301,  2935,  1760,   660,  120;
  24343, 42396, 30630, 14880, 4680, 720;
  ...
		

Crossrefs

Cf. A072597 (column k=1), A000169(n+1) (row sums).

Programs

Mathematica

nn = 8; np = Exp[NestList[x Exp[#] &, x, nn]]; fp = Exp[Log[1/(1 - NestList[x Exp[#] &, x Exp[x], nn])]];Map[Select[#, # > 0 &] &,Prepend[Table[Range[0, nn]! CoefficientList[Series[(fp[[k + 1]] - fp[[k]])*(np[[k + 1]]) + (fp[[k + 1]])*(np[[k + 1]] - np[[k]]) - (fp[[k + 1]] - fp[[k]]) (np[[k + 1]] - np[[k]]), {x, 0, nn}], x], {k, 1, nn - 1}], Range[0, nn]! CoefficientList[Series[1/(1 - x Exp[x])*Exp[x], {x, 0, nn}], x]] // Transpose] // Grid