A281874 Number of Dyck paths of semilength n with distinct peak heights.

1, 1, 1, 3, 5, 13, 31, 71, 181, 447, 1111, 2799, 7083, 17939, 45563, 115997, 295827, 755275, 1929917, 4935701, 12631111, 32340473, 82837041, 212248769, 543978897, 1394481417, 3575356033, 9168277483, 23512924909, 60306860253, 154689354527, 396809130463

Offset: 0

Author

David Callan, Jan 31 2017

Keywords

• nonn

Comments

a(n) is the number of Dyck paths of length 2n with no two peaks at the same height. A peak is a UD, an up-step U=(1,1) immediately followed by a down-step D=(1,-1).

In the Mathematica recurrence below, a(n,k) is the number of Dyck paths of length 2n with all peaks at distinct heights except that there are k peaks at the maximum peak height. Thus a(n)=a(n,1). The recurrence is based on the following simple observation. Paths counted by a(n,k) are obtained from paths counted by a(n-k,i) for some i, 1<=i<=k+1, by inserting runs of one or more contiguous peaks at each of the existing peak vertices at the maximum peak height, except that (at most) one such existing peak may be left undisturbed, and so that a total of k new peaks are added.

It appears that lim a(n)/a(n-1) as n approaches infinity exists and is approximately 2.5659398.

Examples

			a(3)=3 counts UUUDDD, UDUUDD, UUDDUD because the first has only one peak and the last two have peak heights 1,2 and 2,1 respectively.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 19.

Crossrefs

A048285 counts Dyck paths with nondecreasing peak heights.

Column k=1 of A287847, A288108.

Cf. A287846, A287901, A289020.

Programs

Mathematica

a[n_, k_] /; k == n := 1;
a[n_, k_] /; (k > n || k < 1) := 0;
a[n_, k_] :=
a[n, k] =
  Sum[(Binomial[k - 1, i - 1] + i Binomial[k - 1, i - 2]) a[n - k,
     i], {i, k + 1}];
Table[a[n, 1], {n, 28}]