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User: Sanjay Ramassamy

Sanjay Ramassamy has authored 2 sequences.

A350280 Number of bracelets describing topological configurations of points and lines formed by the perpendicular bisectors of the sides of a convex cyclic n-gon.

0, 1, 1, 5, 9, 30, 69, 203, 519, 1466, 3933, 11025, 30345, 85190, 238063, 671651, 1895265, 5376856, 15279117, 43568435, 124478129, 356537150, 1023113061, 2941713513, 8472215013, 24439992746, 70604898953, 204253079165, 591631927785, 1715743930880, 4981202429973

Offset: 1

Sanjay Ramassamy, Dec 23 2021

nonn

For n>=3, a(n) is the number of topological configurations (up to cyclic shifts and reversal) of n points and n lines, where the points lie at the vertices of a convex cyclic n-gon and the lines are the perpendicular bisectors of its sides. Counting such configurations without quotienting out by cyclic shifts and reversal gives the sequence A028243.

a(n) is also the number of equivalence classes (up to cyclic shifts and reversal) of 2n-tuples composed of n 0's and n 1's which have an interlacing signature. The signature of a 2n-tuple (v_1,...,v_{2n}) is the n-tuple (s_1,...,s_n) defined by s_i=v_i+v_{i+n}. The signature is called interlacing if after deleting the 1's, there are letters remaining and the remaining 0's and 2's are alternating.

			For n=3, drawing the three perpendicular bisectors of a triangle divides the plane into 6 regions. Three of these regions contain one vertex of the triangle and the other three contain none. Up to cyclic shifts and reversal, the only possible configuration is (nonempty, nonempty, empty, empty, nonempty, empty), thus a(3)=1.
For n=3, the only 6-tuple (up to cyclic shift and reversal) which has interlacing signature is (1,1,0,0,1,0). Its signature is (1,2,0).
For n=4, the a(4)=5 equivalence classes of 8-tuples with interlacing signature are (0,1,0,1,0,1,0,1), (0,0,0,1,0,1,1,1), (0,1,0,1,0,0,1,1), (0,1,1,1,0,0,1,0) and (0,0,1,1,0,1,1,0).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
P. Melotti, S. Ramassamy and P. Thévenin, Points and lines configurations for perpendicular bisectors of convex cyclic polygons, arXiv:2003.11006 [math.CO], 2020.

Cf. A028243.

\\ here c(n) is up to rotations only.
c(n)={(n%2==0) + sumdiv(n, d, if(n/d%2==1, eulerphi(n/d)*((3^d - (-1)^d)/2 - 2^d)))/n}
a(n)={(c(n) + if(n%2==0, 3^(n/2-1)))/2} \\ Andrew Howroyd, Dec 25 2021

seq(n)=Vec((x^2/(1-x^2) + x^2/(1-3*x^2))/2 + sum(k=0, (n-1)\2, my(d=2*k+1); eulerphi(d)*log((1+x^d)*(1-2*x^d)^2/(1-3*x^d) + O(x*x^n))/d)/4, -n) \\ Andrew Howroyd, Dec 25 2021

From Andrew Howroyd, Dec 25 2021: (Start)
a(n) = (b(n) + (Sum_{d|n, n/d==1 (mod 2)} phi(n/d)*((3^d - (-1)^d)/2 - 2^d))/n)/2 where b(n) = 1 + 3^(n/2-1) for even n and 0 otherwise.
G.f.: (1/2)*(x^2/(1-x^2) + x^2/(1-3*x^2)) + (1/4)*Sum_{k>=0} phi(2*k+1)*log(B(x^(2*k+1)))/(2*k+1) where B(x) = (1+x)*(1-2*x)^2/(1-3*x).
(End)

Terms a(11) and beyond from Andrew Howroyd, Dec 25 2021

A321309 Coefficients of the power series expansion at p=1 of the growth rate C(p) of the length of the longest increasing path in an Erdös-Rényi graph with edge probability p.

Original entry on oeis.org

1, 1, 1, 3, 7, 15, 29, 54, 102, 197, 375, 687, 1226, 2182, 3885, 6828, 11767, 19971, 33519, 55525, 90293, 143350, 221149, 329472, 467362, 611441, 683794, 487644, -425932, -3026915, -9327152, -23364105, -53026834, -113415526, -232986460, -464621237, -905199293

Offset: 0

Sanjay Ramassamy, Nov 03 2018

sign

The entries are known to be integers, they were conjectured to be nonnegative and increasing starting from index 2. The radius of convergence of the generating function is at least (sqrt(2)-1)/2 and at most 1.

C(p) is also the speed of the front of the infinite-bin model with moves following a geometric distribution of parameter p.

			C(1+x) = 1 + x + x^2 + 3x^3 + 7x^4 + 15x^5 + ...

Benjamin Terlat, Table of n, a(n) for n = 0..44
Sergey Foss and Takis Konstantopoulos, Extended renovation theory and limit theorems for stochastic ordered graphs, Markov Process and Related Fields, 9-3 (2003), 413-468.
Sergey Foss, Takis Konstantopoulos, Bastien Mallein, and Sanjay Ramassamy, Last passage percolation and limit theorems in Barak-Erdős directed random graphs and related models, arXiv:2312.02884 [math.PR], 2023. See page 30.
B. Mallein and S. Ramassamy, Barak-Erdös graphs and the infinite-bin model, arXiv:1610.04043 [math.PR], 2016.

Cf. A373089, A373090, A373091.

a(17)-a(20) from Bastien Mallein added by Stefano Spezia, Dec 20 2023

a(21) and beyond from Benjamin Terlat, Jun 24 2024

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User: Sanjay Ramassamy

A350280 Number of bracelets describing topological configurations of points and lines formed by the perpendicular bisectors of the sides of a convex cyclic n-gon.

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