|
%I M4239 N1771 #29 Jul 19 2018 09:49:21
%S 0,0,1,6,41,293,2309,19975,189524,1960041,21993884,266361634,
%T 3465832370,48245601976,715756932697,11277786883720,188135296651083,
%U 3313338641692957,61444453534759589,1196988740015236617,24442368179977776766,522124104504306695929
%N Number of permutations of [n] in which the longest increasing run has length 3.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1. (Values for n>=16 are incorrect.)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000402/b000402.txt">Table of n, a(n) for n = 1..452</a> (first 100 terms from Max Alekseyev)
%H Max A. Alekseyev, <a href="http://arxiv.org/abs/1205.4581">On the number of permutations with bounded runs length</a>, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Oct 23 2012
%e a(4)=6 because we have (124)3, (134)2, (234)1, 4(123), 3(124) and 2(134), where the parentheses surround increasing runs of length 3.
%t b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];
%t T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1];
%t a[n_] := T[n, 3];
%t Array[a, 30] (* _Jean-François Alcover_, Jul 19 2018, after _Alois P. Heinz_ *)
%Y Column 3 of A008304. Other columns: A000303, A000434, A000456, A000467.
%Y Cf. A001250, A001251, A001252, A001253, A010026, A211318.
%K nonn
%O 1,4
%A _N. J. A. Sloane_
%E Better description from _Emeric Deutsch_, May 08 2004
%E Terms a(16), a(17) are corrected and further terms added by _Max Alekseyev_, May 20 2012
|
