A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

1, 1, 4, 26, 1, 237, 19, 2789, 336, 40270, 5981, 405, 689450, 115193, 18900, 13657756, 2459955, 659505, 307348641, 58366045, 20330163, 1375640, 7745565616, 1530739594, 623758590, 99936200, 216114310994, 44076571672, 19795671225, 5325116720

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-largest component is defined to have size 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      26,     1;
     237,    19;
    2789,   336;
   40270,   5981,   405;
  689450, 115193, 18900;
  ...
		

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, A350081, 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^l[1], add(g(i)*
      b(n-i, sort([l[], i])[-3..-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$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[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-3 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0}]}, 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 (4 rows) from Alois P. Heinz, Dec 16 2021