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A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).

%I A350275 #25 Feb 17 2022 14:07:20
%S A350275 1,1,4,27,255,1,3094,31,45865,791,803424,20119,16239720,528991,8505,
%T A350275 372076163,14689441,654885,9529560676,435580164,34859160,269819334245,
%U A350275 13846282341,1646054025,8369112382488,471890017358,73811825010,1286223400
%N A350275 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).
%C A350275 An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
%C A350275 If the mapping has no fourth component, then its fourth-largest component is defined to have size 0.
%H A350275 Alois P. Heinz, <a href="/A350275/b350275.txt">Rows n = 0..100, flattened</a>
%H A350275 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 A350275 Triangle begins:
%e A350275        1;
%e A350275        1;
%e A350275        4;
%e A350275       27;
%e A350275      255,     1;
%e A350275     3094,    31;
%e A350275    45865,   791;
%e A350275   803424, 20119;
%e A350275   ...
%p A350275 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350275 b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
%p A350275       b(n-i, sort([l[], i])[-4..-1])*binomial(n-1, i-1), i=1..n))
%p A350275     end:
%p A350275 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
%p A350275 seq(T(n), n=0..14);  # _Alois P. Heinz_, Dec 22 2021
%t A350275 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350275 b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-4 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
%t A350275 T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350275 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A350275 Row sums give A000312.
%Y A350275 Cf. A001865, A350078, A350079, A350080, A350081, A350276.
%K A350275 nonn,tabf
%O A350275 0,3
%A A350275 _Steven Finch_, Dec 22 2021