A350078 - OEIS

%I #31 Feb 17 2022 14:06:15

%S 1,1,3,1,17,10,142,87,27,1569,911,645,21576,11930,10260,2890,355081,

%T 189610,174132,104720,6805296,3543617,3229275,2493288,705740,

%U 148869153,76060087,67843521,60223520,34424208,3660215680,1842497914,1605373560,1530575960,1051155000,310181886

%N Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose second-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/2).

%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-largest component is defined to have size 0.

%H Alois P. Heinz, <a href="/A350078/b350078.txt">Rows n = 0..140, 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,     10;

%e      142,     87,     27;

%e     1569,    911,    645;

%e    21576,  11930,  10260,   2890;

%e   355081, 189610, 174132, 104720;

%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^l[1], add(g(i)*

%p       b(n-i, sort([l[], i])[-2..-1])*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, [0$2])):

%p seq(T(n), n=0..10);  # _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_] := g[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-2 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];

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

%t Table[T[n], {n, 0, 10}] // 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, A350079, A350080, A350081, A350275, A350276.

%K nonn,tabf

%O 0,3

%A _Steven Finch_, Dec 12 2021

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