A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).

1, 1, 4, 27, 255, 1, 3094, 31, 45865, 791, 803424, 20119, 16239720, 528991, 8505, 372076163, 14689441, 654885, 9529560676, 435580164, 34859160, 269819334245, 13846282341, 1646054025, 8369112382488, 471890017358, 73811825010, 1286223400

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-largest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      27;
     255,     1;
    3094,    31;
   45865,   791;
  803424, 20119;
  ...
		

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, 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^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-4..-1])*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14);  # 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[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-4 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)