

A129163


Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and pyramid weight k.


1



1, 1, 2, 2, 4, 4, 4, 11, 13, 8, 8, 29, 46, 38, 16, 16, 74, 150, 167, 104, 32, 32, 184, 461, 652, 554, 272, 64, 64, 448, 1354, 2344, 2535, 1724, 688, 128, 128, 1072, 3836, 7922, 10462, 9103, 5112, 1696, 256, 256, 2528, 10552, 25506, 40007, 42547, 30773, 14592
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OFFSET

1,3


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, consists of steps U=(1,1)(up), D=(1,1)(down) and L=(1,1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a skew 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 skew Dyck path (word) is the sum of the heights of its maximal pyramids.
Row sums yield A002212. T(n,1)=2^(n2) (n>=2). T(n,n)=2^(n1). Sum(k*T(n,k),k=1..n)=A129164(n). Pyramid weight in Dyck paths is considered in the Denise and Simion reference (see also A091866).


LINKS

Table of n, a(n) for n=1..53.
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155176.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

G.f.=G1, where G=G(t,z) is given by z(1tz)G^2(12tz+tz^2)G+(1z)(1tz)=0.


EXAMPLE

T(3,2)=4 because we have (UD)U(UD)L, U(UD)(UD)D, U(UD)(UD)L and U(UUDD)L (the maximal pyramids are shown between parentheses).
Triangle starts:
1;
1,2;
2,4,4;
4,11,13,8;
8,29,46,38,16;


MAPLE

eq:=z*(1t*z)*G^2(12*t*z+t*z^2)*G+(1z)*(1t*z)=0: G:=RootOf(eq, G): Gser:=simplify(series(G1, z=0, 15)): for n from 1 to 11 do P[n]:=sort(expand(coeff(Gser, z, n))) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form


CROSSREFS

Cf. A002212, A129164.
Sequence in context: A220461 A220287 A320195 * A083549 A083548 A082849
Adjacent sequences: A129160 A129161 A129162 * A129164 A129165 A129166


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Apr 03 2007


STATUS

approved



