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Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).
6

%I #34 Feb 17 2022 14:06:40

%S 1,1,3,1,17,1,9,142,19,27,68,1569,201,135,510,710,21576,2921,3465,

%T 2890,6390,9414,355081,50233,63630,20230,84490,98847,151032,6805296,

%U 1004599,1196181,918680,705740,1493688,1812384,2840648,148869153,22872097,26904339,23943752,6351660,28072548,30810528,38348748,61247664

%N Triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-1).

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

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

%H Alois P. Heinz, <a href="/A350079/b350079.txt">Rows n = 0..141, 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 3, 1;

%e 17, 1, 9;

%e 142, 19, 27, 68;

%e 1569, 201, 135, 510, 710;

%e 21576, 2921, 3465, 2890, 6390, 9414;

%e 355081, 50233, 63630, 20230, 84490, 98847, 151032;

%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)[2],

%p add(b(n-i, sort([l[], i])[1..2])*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$2])):

%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)[[2]], Sum[b[n - i, Sort[Append[l, i]][[1;;2]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];

%t T[n_] := With[{p = b[n, {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 Column 0 gives gives 1 together with A001865.

%Y Row sums give A000312.

%Y Cf. A001865, A350078, A350080, A350081, A350275, A350276

%K nonn,tabf

%O 0,3

%A _Steven Finch_, Dec 12 2021

%E More terms (two rows) from _Alois P. Heinz_, Dec 15 2021