A350276 - OEIS

%I #22 Feb 17 2022 14:09:03

%S 1,1,4,27,255,1,3094,1,30,45865,46,405,340,803424,659,3780,10710,4970,

%T 16239720,12867,48405,209440,178920,87864,372076163,284785,1225665,

%U 3005940,5457060,3558492,1812384,9529560676,7126384,32262300,51205700,135084600,120593340,81557280,42609720

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

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

%H Alois P. Heinz, <a href="/A350276/b350276.txt">Rows n = 0..100, 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       27;

%e      255,   1;

%e     3094,   1,   30;

%e    45865,  46,  405,   340;

%e   803424, 659, 3780, 10710, 4970;

%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)[4],

%p       add(b(n-i, sort([l[], i])[1..4])*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$4])):

%p seq(T(n), n=0..12);  # _Alois P. Heinz_, Dec 22 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)[[4]], Sum[b[n - i, Sort[Append[l, i]][[1 ;; 4]]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];

%t T[n_] := With[{p = b[n, Table[Infinity, {4}]]}, 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, A350081, A350275.

%K nonn,tabf

%O 0,3

%A _Steven Finch_, Dec 22 2021