A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

1, 1, 4, 27, 255, 1, 3094, 1, 30, 45865, 46, 405, 340, 803424, 659, 3780, 10710, 4970, 16239720, 12867, 48405, 209440, 178920, 87864, 372076163, 284785, 1225665, 3005940, 5457060, 3558492, 1812384, 9529560676, 7126384, 32262300, 51205700, 135084600, 120593340, 81557280, 42609720

Offset: 0

Author

Steven Finch, Dec 22 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 fourth component, then its fourth-smallest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      27;
     255,   1;
    3094,   1,   30;
   45865,  46,  405,   340;
  803424, 659, 3780, 10710, 4970;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..100, 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, A350081, A350275.

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)[4],
      add(b(n-i, sort([l[], i])[1..4])*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$4])):
seq(T(n), n=0..12);  # Alois P. Heinz, Dec 22 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)[[4]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 4]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, Table[Infinity, {4}]]}, 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 *)