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 1 to 14 do P[n]:=sort(expand(n!*coeff(Gser, x^n))) od: 1, seq(seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)), n=1..14);

Sequence in context: A121755 A104174 A050400 this_sequence A055584 A066480 A136394

Adjacent sequences: A008968 A008969 A008970 this_sequence A008972 A008973 A008974

tabf,nonn

njas

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