A145901 Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.
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
Offset: 0
Examples
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 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 f-vector 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)
Links
- Sandrine Dasse-Hartaut and Pawel Hitczenko, Greek letters in random staircase tableaux arXiv:1202.3092v1 [math.CO], 2012.
- Peter Bala, Deformations of the Hadamard product of power series
- M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
- S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008.
- 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.
- Gábor Hetyei, The type B permutohedron and the poset of intervals as a Tchebyshev transform, University of North Carolina-Charlotte (2019).
- Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
- Wikipedia, Truncated cuboctahedron
Crossrefs
Programs
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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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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 (* Jean-François Alcover, Jun 02 2019 *)
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).
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)
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 n-th 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*n-1} (-1)^k*T(2*n - 1,k)/((k + 1)*(k + 2)) = -1/2 * 2^(2*n)* Bernoulli(2*n,1/2).
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 n-th 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..n-1} binomial(n,k)2^(n-k)*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)
From Peter Bala, Jan 12 2018: (Start)
n-th 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 k-th row polynomial of A019538. (End)
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