2,5

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids. Row sums are the Fine numbers (A000957). T(n,n)=fibonacci(n-1) (A000045).

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176).

G.f.= G-1, where G=G(t, z) satisfies z(1+t-t^2*z-t^3*z^2)G^2-(1+z-2t^2*z^2)G+1-tz=0.

T(4,3)=3 because we have U(UD)(UUDD)D, U(UUDD)(UD)D and U(UD)(UD)(UD)D, where U=(1,1),D=(1,-1) (the maximal pyramids are shown between parentheses).

Triangle begins:

1;

1,1;

1,3,2;

1,5,9,3;

1,7,21,23,5;

1,9,38,74,56,8;

G:=(1+z-2*t^2*z^2-sqrt((1-z)*(1-z-4*t*z+4*t^2*z^2)))/2/z/(1+t-t^2*z-t^3*z^2)-1: Gser:=simplify(series(G, z=0, 15)): for n from 2 to 13 do P[n]:=expand(coeff(Gser, z^n)) od: for n from 2 to 13 do seq(coeff(P[n], t^j), j=2..n) od; # yields sequence in triangular form

Cf. A000957, A091866, A000045.

Sequence in context: A077951 A077976 A021912 this_sequence A050165 A033878 A144061

Adjacent sequences: A114594 A114595 A114596 this_sequence A114598 A114599 A114600

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 12 2005