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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).

%I A350081 #34 Feb 17 2022 14:08:13
%S A350081 1,1,4,26,1,237,1,18,2789,31,135,170,40270,386,810,3060,2130,689450,
%T A350081 6574,13545,36295,44730,32949,13657756,129291,327285,323680,944300,
%U A350081 790776,604128,307348641,2910709,7207137,6602120,15476580,18780930,16311456,12782916
%N 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).
%C A350081 An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
%C A350081 If the mapping has no third component, then its third-smallest component is defined to have size 0.
%H A350081 Alois P. Heinz, <a href="/A350081/b350081.txt">Rows n = 0..120, flattened</a>
%H A350081 Steven Finch, <a href="http://arxiv.org/abs/2202.07621">Second best, Third worst, Fourth in line</a>, arxiv:2202.07621 [math.CO], 2022.
%e A350081 Triangle begins:
%e A350081        1;
%e A350081        1;
%e A350081        4;
%e A350081       26,    1;
%e A350081      237,    1,    18;
%e A350081     2789,   31,   135,   170;
%e A350081    40270,  386,   810,  3060,  2130;
%e A350081   689450, 6574, 13545, 36295, 44730, 32949;
%e A350081   ...
%p A350081 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350081 b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],
%p A350081       add(b(n-i, sort([l[], i])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))
%p A350081     end:
%p A350081 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$3])):
%p A350081 seq(T(n), n=0..12);  # _Alois P. Heinz_, Dec 17 2021
%t A350081 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350081 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 A350081 T[n_] := With[{p = b[n, {Infinity, Infinity, Infinity}]}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350081 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A350081 Row sums give A000312.
%Y A350081 Cf. A001865, A350078, A350079, A350080, A350275, A350276.
%K A350081 nonn,tabf
%O A350081 0,3
%A A350081 _Steven Finch_, Dec 12 2021
%E A350081 More terms (two rows) from _Alois P. Heinz_, Dec 16 2021