0,6

Row n has 1+floor(n/2) terms.

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

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

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).

Triangle starts

1;

1;

1,1;

1,5;

1,18,5;

1,58,61;

Example: T(3,1)=5 because all permutations of [3] with the exception of 321 have one increasing run of length at least 2.

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);

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start)

A160486 is a sub-triangle.

A000340, A000363, A000507 equal the second, third and fourth left hand columns.

(End)

Sequence in context: A121755 A104174 A050400 this_sequence A151335 A055584 A146055

Adjacent sequences: A008968 A008969 A008970 this_sequence A008972 A008973 A008974

tabf,nonn

N. J. A. Sloane (njas(AT)research.att.com).

Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu) and Ira Gessel (gessel(AT)brandeis.edu), Aug 28 2004

Maple program edited by Johannes W. Meijer (meijgia(AT)hotmail.com), May 15 2009