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.

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

0,4

Comments

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.

Links

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.

Example

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

Crossrefs

Cf. A373089, A373090, A373091.

Sequence in context: A122768 A344743 A373091 * A373090 A023608 A218189

Adjacent sequences: A321306 A321307 A321308 * A321310 A321311 A321312

Keyword

sign

Author

Sanjay Ramassamy, Nov 03 2018

Extensions

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

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

Status

approved