

A145901


Triangle of fvectors of the simplicial complexes dual to the permutohedra of type B_n.


20



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
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OFFSET

0,3


COMMENTS

The Coxeter group of type B_n may be realized as the group of n X n matrices with exactly one nonzero 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 fvectors 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 hvectors of type B permutohedra.
This is the (unsigned) triangle of connection constants between the polynomial sequences (2*x + 1)^n, n >= 0, and binomial(x+k,k), k >= 0. For example, (2*x + 1)^2 = 8*binomial(x+2,2)  8*binomial(x+1,1) + 1 and (2*x + 1)^3 = 48*binomial(x+3,3)  72*binomial(x+2,2) + 26*binomial(x+1,1)  1. Cf. A163626.  Peter Bala, Jun 06 2019


LINKS

Table of n, a(n) for n=0..41.
Sandrine DasseHartaut and Pawel Hitczenko, Greek letters in random staircase tableaux arXiv:1202.3092v1 [math.CO], 2012.
P. Bala, Deformations of the Hadamard product of power series
M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/ParkCity 2004; arXiv:math/0505518 [math.CO], 20052008.
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.
ShiMei Ma, A family of twovariable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
Wikipedia, Truncated cuboctahedron


FORMULA

T(n,k) = Sum_{i = 0..k} (1)^(ki)*binomial(k,i)*(2*i+1)^n.
Recurrence relation: T(n,k) = (2*k + 1)*T(n1,k) + 2*k*T(n1,k1) 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)*It*A062715)^(1) are, apart from the initial 1, the row polynomials of this table with an extra factor of t. Cf. A060187. (End)
From Peter Bala, Jul 18 2013: (Start)
Integrating the above e.g.f. with respect to x from x = 0 to x = 1 gives Sum_{k = 0..n} (1)^k*T(n,k)/(k + 1) = 2^n*Bernoulli(n,1/2), the nth cosecant number.
The corresponding Type A result is considered in A028246 as Worpitzky's algorithm.
Also for n >= 0, Sum_{k = 0..2*n} (1)^k*T(2*n,k)/((k + 1)*(k + 2)) = 1/2*2^(2*n)*Bernoulli(2*n,1/2) and for n >= 1, Sum_{k = 0..2*n1} (1)^k*T(2*n  1,k)/((k + 1)*(k + 2)) = 1/2 * 2^(2*n)* Bernoulli(2*n,1/2).
The nonzero cosecant numbers are given by A001896/A001897. (End)
From Peter Bala, Jul 22 2014: (Start)
The row polynomials R(n,x) satisfy the recurrence equation R(n+1,x) = D(R(n,x)) with R(0,x) = 1, where D is the operator 1 + 2*x + 2*x(1 + x)*d/dx.
R(n,x) = 1/(1 + x)* Sum_{k = 0..inf} (2*k + 1)^n*(x/(1 + x))^k, valid for x in the open interval (1/2, inf). Cf. A019538.
The shifted row polynomial x*R(n,x) = (1 + x)^n*P(n,x/(1 + x)) where P(n,x) denotes the nth row polynomial of A060187.
The row polynomials R(n,x) have only real zeros.
Symmetry: R(n,x) = (1)^n*R(n,1  x). Consequently the zeros of R(n,x) lie in the open interval (1, 0). (End)
From Peter Bala, May 28 2015: (Start)
Recurrence for row polynomials: R(n,x) = 1 + x*Sum_{k = 0..n1} binomial(n,k)2^(nk)*R(k,x) with R(0,x) = 1.
For a fixed integer k, the expansion of the function A(k,z) := exp( Sum_{n >= 1} R(n,k)*z^n/n ) has integer coefficients and satisfies the functional equation A(k,z)^(k + 1) = 1/(1  z)*( BINOMIAL(BINOMIAL(A(k,z))) )^k, where BINOMIAL(F(z))= 1/(1  z)*F(z/(1  z)) denotes the binomial transform of the o.g.f. F(z). A(k,z) = A((k + 1),z). Cf. A019538.
For cases see A258377 (k = 1), A258378(k = 2), A258379 (k = 3), A258380 (k = 4) and A258381 (k = 5). (End)
T(n,k) = A154537(n,k)*k! = A039755(n,k)*(2^k*k!), 0 <= k <= n.  Wolfdieter Lang, Apr 19 2017
From Peter Bala, Jan 12 2018: (Start)
nth row polynomial R(n,x) = (1 + 2*x) o (1 + 2*x) o ... o (1 + 2*x) (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See example E13 in the Bala link.
R(n,x) = Sum_{k = 0..n} binomial(n,k)*2^k*F(k,x) where F(k,x) is the Fubini polynomial of order k, the kth row polynomial of A019538. (End)


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 fvector (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 realized 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 fvector is (1,26,72,48)  row 4 of this triangle.
From Peter Bala, Jun 06 2019: (Start)
Examples of falling factorials identities for odd numbered rows: Let (x)_n = x*(x  1)*...*(x  n + 1) with (x)_0 = 1 denote the falling factorial power.
Row 1: 2*(x)_1 + (0  2*x)_1 = 0.
Row 3: 48*(x)_3 + 72*(x)_2 * (2  2*x)_1 + 26*(x)_1 * (2  2*x)_2 + (2  2*x)_3 = 0
Row 5: 3840*(x)_5 + 9600*(x)_4 * (4  2*x)_(1) + 8160*(x)_3 * (4  2*x)_2 + 2640*(x)_2 * (4  2*x)_3 + 242*(x)_1 * (4  2*x)_4 + (4  2*x)_5 = 0. (End)


MAPLE

with(combinat):
T:= (n, k) > add((1)^(ki)*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;


MATHEMATICA

T[n_, k_] := Sum[(1)^(k  i)*Binomial[k, i]*(2*i + 1)^n, {i, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Jun 02 2019 *)


CROSSREFS

Cf. A019538 (fvectors type A permutohedra), A060187 (hvectors type B permutohedra), A080253 (row sums), A145905, A062715, A028246.
Cf. A258377, A258378, A258379, A258380, A258381.
Cf. A039755, A154537.
Sequence in context: A075733 A127674 A271316 * A321369 A286724 A123516
Adjacent sequences: A145898 A145899 A145900 * A145902 A145903 A145904


KEYWORD

easy,nonn,tabl


AUTHOR

Peter Bala, Oct 26 2008


STATUS

approved



