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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).
6

%I #34 Feb 17 2022 14:08:13

%S 1,1,4,26,1,237,1,18,2789,31,135,170,40270,386,810,3060,2130,689450,

%T 6574,13545,36295,44730,32949,13657756,129291,327285,323680,944300,

%U 790776,604128,307348641,2910709,7207137,6602120,15476580,18780930,16311456,12782916

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

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

%H Alois P. Heinz, <a href="/A350081/b350081.txt">Rows n = 0..120, flattened</a>

%H 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 Triangle begins:

%e 1;

%e 1;

%e 4;

%e 26, 1;

%e 237, 1, 18;

%e 2789, 31, 135, 170;

%e 40270, 386, 810, 3060, 2130;

%e 689450, 6574, 13545, 36295, 44730, 32949;

%e ...

%p g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:

%p b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[3],

%p add(b(n-i, sort([l[], i])[1..3])*g(i)*binomial(n-1, i-1), i=1..n))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$3])):

%p seq(T(n), n=0..12); # _Alois P. Heinz_, Dec 17 2021

%t g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];

%t 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 T[n_] := With[{p = b[n, {Infinity, Infinity, Infinity}]}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)

%Y Row sums give A000312.

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

%K nonn,tabf

%O 0,3

%A _Steven Finch_, Dec 12 2021

%E More terms (two rows) from _Alois P. Heinz_, Dec 16 2021