A066094 Type D Eulerian triangle.

1, 1, 1, 1, 2, 1, 1, 11, 11, 1, 1, 44, 102, 44, 1, 1, 157, 802, 802, 157, 1, 1, 530, 5551, 10876, 5551, 530, 1, 1, 1731, 35121, 124427, 124427, 35121, 1731, 1, 1, 5528, 208732, 1265704, 2201030, 1265704, 208732, 5528, 1

Offset: 0

Author

Paul Boddington, Mar 05 2003

Keywords

• easy
• nonn
• tabl

Comments

Let n >= 2 and write the polynomial D(n,0)+D(n,1)*x+...+D(n,n)*x^n as a polynomial in y := x-1. Then the coefficient of y^r is the number of cells of dimension n-r in the cellular decomposition of a Euclidean space containing a root system of type D_n. If n >= 2 then the corresponding row sum is 2^(n-1)*n!, while Sum_{k=0..n} 2^k*D(n,k) is given by sequence A080254. [Row sum formula corrected by Joshua Swanson, Jul 12 2022]

The entries in row n (for n >= 2) are the components of the h-vector of the permutohedra of type D_n. See A145902 for the corresponding array of f-vectors for type D permutohedra. [Peter Bala, Oct 29 2008]

Examples

			From _Peter Bala_, Oct 29 2008: (Start)
The triangle begins
n\k|..0....1....2....3....4....5
================================
0..|..1
1..|..1....1
2..|..1....2....1
3..|..1...11...11....1
4..|..1...44..102...44....1
5..|..1..157..802..802..157....1
...
(End)
		

References

• K. S. Brown, Buildings, Springer-Verlag, 1988
• T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 11.

Links

• Jose Bastidas, Table of n, a(n) for n = 0..1274 (First 50 rows)
• Anna Borowiec and Wojciech Mlotkowski, New Eulerian numbers of type D, arXiv:1509.03758 [math.CO], 2015.
• Chak-On Chow, On the Eulerian polynomials of type D, arXiv:math/0201140 [math.CO], 2002.
• Nolan J. Coble and Alexander Barg, Classical and quantum Coxeter codes: Extending the Reed-Muller family, arXiv:2502.14746 [cs.IT], 2025. See p. 7.
• Shi-Mei Ma, Polynomials with only real zeros and the Eulerian polynomials of type D, arXiv preprint arXiv:1205.6242 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 23 2012

Crossrefs

Cf. A008292, A060187, A080254.

Cf. A145902. [Peter Bala, Oct 29 2008]

Formula

Let D(n, k) denote the (k+1)st entry in the (n+1)st row and let A(n, k), B(n, k) be triangles A008292 (The Eulerian triangle), A060187 respectively. Then D(n, k) = B(n, k)-2^(n-1)*n*A(n-2, k-1).

Chow gives complicated recurrences and generating functions.

E.g.f.: [(1-x)*exp(z*(1-x)) - z*x*(1-x)*exp(2*z*(1-x))]/(1 - x*exp(2*z*(1-x))) = 1 + x*z + (1 + 2*x + x^2)*z^2/2! + (1 + 11*x + 11*x^2 + x^3)*z^3/3! + ... . [Peter Bala, Oct 29 2008]