0,3

The Coxeter group of type B_n may be realised as the group of n x n matrices with exactly one non-zero entry in each row and column, that entry being either +1 or -1. The order of the group is 2^n*n!. The orbit of the point (1,2,...,n) (or any sufficiently generic point (x_1,...,x_n)) under the action of this group is a set of 2^n*n! distinct points whose convex hull is defined to be the permutohedron of type B_n. The rows of this table are the f-vectors of the simplicial complexes dual to these type B permutohedra. Some examples are given in the Example section below. See A060187 for the corresponding table of h-vectors of type B permutohedra.

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004.

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

Wikipedia entry, Truncated cuboctahedron

T(n,k) = sum {i = 0..k} (-1)^(k-i)*binomial(k,i)*(2*i+1)^n. Recurrence relation: T(n,k) = (2*k + 1)*T(n-1,k) + 2*k*T(n-1,k-1) with T(0,0) = 1 and T(0,k) = 0 for k >= 1. Relation with type B Stirling numbers of the second kind: T(n,k) = 2^k*k!*A039755(n,k). Row sums A080253. The matrix product A060187 * A007318 produces the mirror image of this triangle.

E.g.f.: exp(t)/(1 + x - x* exp(2*t)) = 1 + (1 + 2*x)*t + (1 + 8*x + 8*x^2 )*t^2/2! + ... .

The triangle begins

n\k|..0.....1.....2.....3.....4.....5

=====================================

0..|..1

1..|..1.....2

2..|..1.....8.....8

3..|..1....26....72....48

4..|..1....80...464...768...384

5..|..1...242..2640..8160..9600..3840

...

Row 2: the permutohedron of type B_2 is an octagon with 8 vertices and 8 edges. Its dual, also an octagon, has f-vector (1,8,8) - row 3 of this triangle.

Row 3: for an appropriate choice of generic point in R_3, the permutohedron of type B_3 is realised as the great rhombicuboctahedron, also known as the truncated cuboctahedron, with 48 vertices, 72 edges and 26 faces (12 squares, 8 regular hexagons and 6 regular octagons). See the Wikipedia entry and also [Fomin and Reading p.22]. Its dual polyhedron is a simplicial polyhedron, the disdyakis dodecahedron, with 26 vertices, 72 edges and 48 triangular faces and so its f-vector is (1,26,72,48) - row 4 of this triangle.

with(combinat):

T:= (n, k) -> add((-1)^(k-i)*binomial(k, i)*(2*i+1)^n, i = 0..k):

for n from 0 to 9 do

seq(T(n, k), k = 0..n);

end do;

A019538 (f-vectors type A permutohedra), A060187(h-vectors type B permutohedra), A080253 (row sums), A145905.

Sequence in context: A021461 A075733 A127674 this_sequence A123516 A016446 A086657

Adjacent sequences: A145898 A145899 A145900 this_sequence A145902 A145903 A145904

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008

Typo corrected by Peter Bala (pbala(AT)talktalk.net), May 31 2009