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A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).

%I A350276 #22 Feb 17 2022 14:09:03
%S A350276 1,1,4,27,255,1,3094,1,30,45865,46,405,340,803424,659,3780,10710,4970,
%T A350276 16239720,12867,48405,209440,178920,87864,372076163,284785,1225665,
%U A350276 3005940,5457060,3558492,1812384,9529560676,7126384,32262300,51205700,135084600,120593340,81557280,42609720
%N A350276 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-smallest component has size exactly k; n >= 0, 0 <= k <= max(0,n-3).
%C A350276 An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
%C A350276 If the mapping has no fourth component, then its fourth-smallest component is defined to have size 0.
%H A350276 Alois P. Heinz, <a href="/A350276/b350276.txt">Rows n = 0..100, flattened</a>
%H A350276 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 A350276 Triangle begins:
%e A350276        1;
%e A350276        1;
%e A350276        4;
%e A350276       27;
%e A350276      255,   1;
%e A350276     3094,   1,   30;
%e A350276    45865,  46,  405,   340;
%e A350276   803424, 659, 3780, 10710, 4970;
%e A350276   ...
%p A350276 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350276 b:= proc(n, l) option remember; `if`(n=0, x^subs(infinity=0, l)[4],
%p A350276       add(b(n-i, sort([l[], i])[1..4])*g(i)*binomial(n-1, i-1), i=1..n))
%p A350276     end:
%p A350276 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [infinity$4])):
%p A350276 seq(T(n), n=0..12);  # _Alois P. Heinz_, Dec 22 2021
%t A350276 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350276 b[n_, l_] := b[n, l] = If[n == 0, x^(l /. Infinity -> 0)[[4]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 4]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
%t A350276 T[n_] := With[{p = b[n, Table[Infinity, {4}]]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350276 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A350276 Row sums give A000312.
%Y A350276 Cf. A001865, A350078, A350079, A350080, A350081, A350275.
%K A350276 nonn,tabf
%O A350276 0,3
%A A350276 _Steven Finch_, Dec 22 2021