A145903 Generalized Narayana numbers for root systems of type D_n. Triangle of h-vectors of type D associahedra.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 24, 12, 1, 1, 20, 70, 70, 20, 1, 1, 30, 165, 280, 165, 30, 1, 1, 42, 336, 875, 875, 336, 42, 1, 1, 56, 616, 2296, 3500, 2296, 616, 56, 1, 1, 72, 1044, 5292, 11466, 11466, 5292, 1044, 72, 1

Offset

0,5

Comments

The generalized Narayana numbers of type D_n (row n of this triangle) are defined as the entries of the h-vector of the simplicial complex dual to the generalized associahedron of type D_n [Fomin & Reading, p.60]. For the corresponding triangle of f-vectors see A080721. For Narayana numbers of root systems of type A and type B see A001263 and A008459 respectively.

References

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.

Links

Table of n, a(n) for n=0..54.

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008.

Formula

For n >= 2, T(n,k) = binomial(n,k)^2 - n/(n-1)*binomial(n-1,k-1)*binomial(n-1,k).

Example

Root systems of type D_n are defined only for n >= 2. It seems convenient to complete the array to form a lower unit triangular matrix.
Triangle starts
n\k|..0....1....2....3....4....5....6
=====================================
0..|..1
1..|..1....1
2..|..1....2....1
3..|..1....6....6....1
4..|..1...12...24...12....1
5..|..1...20...70...70...20....1
6..|..1...30..165..280..165...30....1
...

Crossrefs

A001263, A008459, A051924 (row sums), A080721.

Sequence in context: A157635 A075798 A155864 * A223257 A173881 A329228

Adjacent sequences: A145900 A145901 A145902 * A145904 A145905 A145906

Keyword

easy,nonn,tabl

Author

Peter Bala, Oct 28 2008

Status

approved