A373089 -id:A373089 - cp's OEIS frontend

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

Sanjay Ramassamy, Nov 03 2018

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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

A373090 Coefficients of the power series expansion at p=1 of the time constant C(-2,p) for last passage percolation on the complete directed acyclic graph, where the edges' weights are equal to 1 or -2 with respective probabilities p and 1-p.

Original entry on oeis.org

1, 1, 1, 3, 7, 15, 29, 54, 102, 197, 376, 695, 1260, 2286, 4155, 7489, 13347, 23621, 41609, 72884, 126789, 218903, 375140, 638554, 1079382, 1809256, 3003411, 4934260, 8013764, 12839395, 20232603, 31228335, 46918878, 67947178, 93185004, 116654299, 120921410, 63471736, -150813354, -723950195

Offset: 0

Benjamin Terlat, May 23 2024

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C(-2,p) is also the speed of the front for an interacting particle system with 3 bins, which corresponds to the particular case of the max-growth system where the probability distribution has two atoms 1 and -2 with respective probabilities p and 1-p.

The first 10 coefficients of this sequence coincide with the first 10 coefficients of A321309.

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

Benjamin Terlat, Table of n, a(n) for n = 0..1250
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.
Sergey Foss, Takis Konstantopoulos, Bastien Mallein, and Sanjay Ramassamy, Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation, arXiv:2110.01559 [math.PR], 2023.

Cf. A321309, A373089, A373091.

A373091 Coefficients of the power series expansion at p=1 of the time constant C(-3,p) for last passage percolation on the complete directed acyclic graph, where the edges' weights are equal to 1 or -3 with respective probabilities p and 1-p.

Original entry on oeis.org

1, 1, 1, 3, 7, 15, 29, 54, 102, 197, 375, 687, 1226, 2182, 3885, 6827, 11757, 19920, 33339, 55012, 88980, 140141, 213535, 311997, 428578, 527659, 506451, 118728, -1180673, -4546846, -12344870, -29279209, -64481947, -135339292, -274463246, -542210697, -1048748528, -1992459450

Offset: 0

Benjamin Terlat, May 23 2024

sign

C(-3,p) is also the speed of the front for an interacting particle system with 4 bins, which corresponds to the particular case of the max-growth system where the probability distribution has two atoms 1 and -3 with respective probabilities p and 1-p.

The first 15 coefficients of this sequence coincide with the first 15 coefficients of A321309.

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

Benjamin Terlat, Table of n, a(n) for n = 0..750
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.
Sergey Foss, Takis Konstantopoulos, Bastien Mallein, and Sanjay Ramassamy, Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation, arXiv:2110.01559 [math.PR], 2023.

Cf. A321309, A373089, A373090.

cp's OEIS Frontend

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.

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A373090 Coefficients of the power series expansion at p=1 of the time constant C(-2,p) for last passage percolation on the complete directed acyclic graph, where the edges' weights are equal to 1 or -2 with respective probabilities p and 1-p.

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A373091 Coefficients of the power series expansion at p=1 of the time constant C(-3,p) for last passage percolation on the complete directed acyclic graph, where the edges' weights are equal to 1 or -3 with respective probabilities p and 1-p.

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