%I A008971
%S A008971 1,1,1,1,1,5,1,18,5,1,58,61,1,179,479,61,1,543,3111,1385,1,1636,18270,
%T A008971 19028,1385,1,4916,101166,206276,50521,1,14757,540242,1949762,1073517,
%U A008971 50521,1,44281,2819266,16889786,17460701,2702765,1,132854,14494859
%N A008971 Triangle read by rows: T(n,k) is the number of permutations of [n] with 
               k increasing runs of length at least 2. Triangle starts 1; 1; 1,1; 
               1,5; 1,18,5; 1,58,61; Row n has 1+floor(n/2) terms.
%C A008971 Row n has 1+floor(n/2) terms.
%C A008971 T(n,k) is also the number of permutations of [n] with k "exterior peaks" 
               where we count peaks in the usual way, but add a peak at the beginning 
               if the permutation begins with a descent, e.g. 213 has one exterior 
               peak. T(3,1) = 5 since all permutations of [3] have an exterior peak 
               except 123. This is also the definition for peaks of signed permutations, 
               where we assume that a signed permutation always begins with a zero. 
               - T. Kyle Petersen (tkpeters(AT)brandeis.edu), Jan 14 2005
%D A008971 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, 
               Boca Raton, Florida, 2002.
%D A008971 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 260.
%F A008971 E.g.f. = G(t, x) = sum(T(n, k)t^k x^n/n!, 0<=k<=floor(n/2), n>=0)= sqrt(1-t)/
               [sqrt(1-t)*cosh(xsqrt(1-t))-sinh(xsqrt(1-t))] (Ira Gessel). T(n+1, 
               k)=(2k+1)T(n, k) + (n-2k+2)T(n, k-1) (see p. 542 of the Charalambides 
               reference).
%e A008971 Triangle starts
%e A008971 1;
%e A008971 1;
%e A008971 1,1;
%e A008971 1,5;
%e A008971 1,18,5;
%e A008971 1,58,61;
%e A008971 Example: T(3,1)=5 because all permutations of [3] with the exception 
               of 321 have one increasing run of length at least 2.
%p A008971 G:=sqrt(1-t)/(sqrt(1-t)*cosh(x*sqrt(1-t))-sinh(x*sqrt(1-t))): Gser:=simplify(series(G,
               x=0,15)): for n from 0 to 14 do P[n]:=sort(expand(n!*coeff(Gser,x,
               n))) od: seq(seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)),n=0..14);
%Y A008971 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 
               2009: (Start)
%Y A008971 A160486 is a sub-triangle.
%Y A008971 A000340, A000363, A000507 equal the second, third and fourth left hand 
               columns.
%Y A008971 (End)
%Y A008971 Sequence in context: A121755 A104174 A050400 this_sequence A151335 A055584 
               A146055
%Y A008971 Adjacent sequences: A008968 A008969 A008970 this_sequence A008972 A008973 
               A008974
%K A008971 tabf,nonn
%O A008971 0,6
%A A008971 N. J. A. Sloane (njas(AT)research.att.com).
%E A008971 Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu) and Ira Gessel (gessel(AT)brandeis.edu), 
               Aug 28 2004
%E A008971 Maple program edited by Johannes W. Meijer (meijgia(AT)hotmail.com), 
               May 15 2009