cp's OEIS Frontend

A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).

%I A350080 #28 Feb 17 2022 14:07:03
%S A350080 1,1,4,26,1,237,19,2789,336,40270,5981,405,689450,115193,18900,
%T A350080 13657756,2459955,659505,307348641,58366045,20330163,1375640,
%U A350080 7745565616,1530739594,623758590,99936200,216114310994,44076571672,19795671225,5325116720
%N A350080 Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose third-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/3).
%C A350080 An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
%C A350080 If the mapping has no third component, then its third-largest component is defined to have size 0.
%H A350080 Alois P. Heinz, <a href="/A350080/b350080.txt">Rows n = 0..120, flattened</a>
%H A350080 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 A350080 Triangle begins:
%e A350080        1;
%e A350080        1;
%e A350080        4;
%e A350080       26,     1;
%e A350080      237,    19;
%e A350080     2789,   336;
%e A350080    40270,   5981,   405;
%e A350080   689450, 115193, 18900;
%e A350080   ...
%p A350080 g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
%p A350080 b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
%p A350080       b(n-i, sort([l[], i])[-3..-1])*binomial(n-1, i-1), i=1..n))
%p A350080     end:
%p A350080 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$3])):
%p A350080 seq(T(n), n=0..12);  # _Alois P. Heinz_, Dec 17 2021
%t A350080 g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
%t A350080 b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-3 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
%t A350080 T[n_] := With[{p = b[n, {0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350080 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A350080 Row sums give A000312.
%Y A350080 Cf. A001865, A350078, A350079, A350081, A350275, A350276.
%K A350080 nonn,tabf
%O A350080 0,3
%A A350080 _Steven Finch_, Dec 12 2021
%E A350080 More terms (4 rows) from _Alois P. Heinz_, Dec 16 2021