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A145901
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Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.
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6
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1, 1, 2, 1, 8, 8, 1, 26, 72, 48, 1, 80, 464, 768, 384, 1, 242, 2640, 8160, 9600, 3840, 1, 728, 14168, 72960, 151680, 138240, 46080, 1, 2186, 73752, 595728, 1948800, 3037440, 2257920, 645120, 1, 6560, 377504, 4612608, 22305024, 52899840
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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.
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REFERENCES
| Sandrine Dassehartaut and Pawel Hitczenko, Greek letters in random staircase tableaux, http://www.math.drexel.edu/~phitczen/st_pap_fin.pdf.
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LINKS
| 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
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FORMULA
| 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! + ... .
From Peter Bala, Oct 13 2011: (Start)
The polynomials in the first column of the array ((1+t)*P^(-1)-t*P)^(-1), P Pascal's triangle and I the identity, are the row polynomials of this table.
The polynomials in the first column of the array ((1+t)*I-t*A062715)^(-1) are, apart from the initial 1, the row polynomials of this table with an extra factor of t. Cf. A060187. (End)
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EXAMPLE
| 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.
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MAPLE
| 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;
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CROSSREFS
| A019538 (f-vectors type A permutohedra), A060187(h-vectors type B permutohedra), A080253 (row sums), A145905. A062715.
Sequence in context: A021461 A075733 A127674 * A123516 A193604 A016446
Adjacent sequences: A145898 A145899 A145900 * A145902 A145903 A145904
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008
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EXTENSIONS
| Typo corrected by Peter Bala (pbala(AT)talktalk.net), May 31 2009
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