%I A145901
%S A145901 1,1,2,1,8,8,1,26,72,48,1,80,464,768,384,1,242,2640,8160,9600,3840,1,
%T A145901 728,14168,72960,151680,138240,46080,1,2186,73752,595728,1948800,
%U A145901 3037440,2257920,645120,1,6560,377504,4612608,22305024,52899840
%N A145901 Triangle of f-vectors of the simplicial complexes dual to the permutohedra 
               of type B_n.
%C A145901 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.
%H A145901 S. Fomin, N. Reading, <a href="http://arxiv.org/abs/math.CO/0505518">
               Root systems and generalized associahedra</a>, Lecture notes for 
               IAS/Park-City 2004.
%H A145901 Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               vol9.html">On a Number Pyramid Related to the Binomial, Deleham, 
               Eulerian, MacMahon and Stirling number triangles</a>, Journal of 
               Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H A145901 Wikipedia entry, <a href="http://en.wikipedia.org/wiki/Truncated_cuboctahedron"> 
               Truncated cuboctahedron</a>
%F A145901 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.
%F A145901 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! + ... .
%e A145901 The triangle begins
%e A145901 n\k|..0.....1.....2.....3.....4.....5
%e A145901 =====================================
%e A145901 0..|..1
%e A145901 1..|..1.....2
%e A145901 2..|..1.....8.....8
%e A145901 3..|..1....26....72....48
%e A145901 4..|..1....80...464...768...384
%e A145901 5..|..1...242..2640..8160..9600..3840
%e A145901 ...
%e A145901 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.
%e A145901 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.
%p A145901 with(combinat):
%p A145901 T:= (n,k) -> add((-1)^(k-i)*binomial(k,i)*(2*i+1)^n,i = 0..k):
%p A145901 for n from 0 to 9 do
%p A145901 seq(T(n,k),k = 0..n);
%p A145901 end do;
%Y A145901 A019538 (f-vectors type A permutohedra), A060187(h-vectors type B permutohedra), 
               A080253 (row sums), A145905.
%Y A145901 Sequence in context: A021461 A075733 A127674 this_sequence A123516 A016446 
               A086657
%Y A145901 Adjacent sequences: A145898 A145899 A145900 this_sequence A145902 A145903 
               A145904
%K A145901 easy,nonn,tabl
%O A145901 0,3
%A A145901 Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008
%E A145901 Typo corrected by Peter Bala (pbala(AT)talktalk.net), May 31 2009