A208740 Number of multisets that occurring as the peak heights multiset of a Dyck n-path that are the also the peak heights multiset of a smaller Dyck path.

0, 0, 0, 1, 4, 13, 34, 83, 189, 415, 885, 1853, 3824, 7819, 15876, 32084, 64621, 129860, 260547, 522201, 1045862, 2093646, 4189796, 8382845, 16769878, 33545136, 67097132, 134202986, 268416996, 536847887, 1073713195, 2147448177, 4294923476, 8589880629

Offset

1,5

Comments

We use the definition given by Callan and Deutsch (see reference). A Dyck n-path is a lattice path of n upsteps U (changing by (1,1)) and n downsteps D (changing by (1,-1)) that starts at the origin and never goes below the x-axis. A peak is an occurrence of U D and the peak height is the y-coordinate of the vertex between its U and D.

Also the number of nonempty multisets S of positive integers satisfying max(S) + |S| <= n <= sum(S).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

D. Callan and E. Deutsch, Problems and Solutions: 11624, The Amer. Math. Monthly 119 (2012), no. 2, 161-162.

Formula

a(n) = 2^(n-1) - A000070(n-1).

a(n) = A208738(n) - 2^(n-1).

G.f.: x/(1-2*x)-(x/(1-x))*product(m>=1, 1/(1-x^m)).

Example

For a Dyck 4-path there is only one peak heights multiset occurring also for a Dyck 3-path. This is {2,2} and occurs for both UUDDUUDD when n=4 and UUDUDD when n=3.

Mathematica

Table[2^(n - 1) - Sum[PartitionsP[k], {k, 0, n - 1}], {n, 1, 40}]

Prog

(PARI) a(n) = 2^(n-1) - sum(k=0, n-1, numbpart(k)); \\ Michel Marcus, Jul 07 2018

Crossrefs

Cf. A000070, A208738, A208739.

Sequence in context: A213578 A212149 A357284 * A127981 A296303 A089453

Adjacent sequences: A208737 A208738 A208739 * A208741 A208742 A208743

Keyword

nonn

Author

David Nacin, Mar 01 2012

Status

approved