A350081 Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-2).

1, 1, 4, 26, 1, 237, 1, 18, 2789, 31, 135, 170, 40270, 386, 810, 3060, 2130, 689450, 6574, 13545, 36295, 44730, 32949, 13657756, 129291, 327285, 323680, 944300, 790776, 604128, 307348641, 2910709, 7207137, 6602120, 15476580, 18780930, 16311456, 12782916

Offset: 0

Author

Steven Finch, Dec 12 2021

Keywords

• nonn
• tabf

Comments

An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.

If the mapping has no third component, then its third-smallest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      26,    1;
     237,    1,    18;
    2789,   31,   135,   170;
   40270,  386,   810,  3060,  2130;
  689450, 6574, 13545, 36295, 44730, 32949;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..120, flattened
• Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Crossrefs

Row sums give A000312.

Cf. A001865, A350078, A350079, A350080, A350275, A350276.

Programs

Maple

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],
      add(b(n-i, sort([l[], i])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$3])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 17 2021

Mathematica

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[3]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 3]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {Infinity, Infinity, Infinity}]}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Extensions

More terms (two rows) from Alois P. Heinz, Dec 16 2021