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August  2021, 4(3): 185-191. doi: 10.3934/mfc.2021011

Asymptotic normality of associated Lah numbers

1. 

Shandong Yutai No.1 Middle School, Yutai 272300, China

2. 

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

* Corresponding author: Lily Li Liu

Received  September 2019 Revised  June 2021 Published  August 2021 Early access  August 2021

Fund Project: This work was supported partially by the National Natural Science Foundation of China (No. 11871304) and the Natural Science Foundation of Shandong Province of China (No. ZR2017MA025)

Based on the results given by Ahuja and Enneking, we show that the generating function of the associated Lah numbers having only real zeros, and further obtain the asymptotic normality of the associated Lah numbers. As application, we get the asymptotic normality of the signless Lah numbers.

Citation: Wen Zhang, Lily Li Liu. Asymptotic normality of associated Lah numbers. Mathematical Foundations of Computing, 2021, 4 (3) : 185-191. doi: 10.3934/mfc.2021011
References:
[1]

J. C. Ahuja, Distributions of the Sum of Independent Decapitated Negative Binomial Variables, Ann. Math. Statist., 42 (1971), 383-384.  doi: 10.1214/aoms/1177693527.

[2]

J. C. Ahuja and E. A. Enneking, Concavity property and a recurrence relation for associated Lah nubers, Fibonacci Quart., 17 (1979), 158-161. 

[3]

P. Baldi and Y. Rinott, Asymptotic normality of some graph-related statistics, J. Appl. Probab., 26 (1989), 171-175.  doi: 10.2307/3214327.

[4]

P. Barry, Some observations on the Lah and Laguerre transforms of integer sequences, J. Integer Seq., 10 (2007), 18pp.

[5]

H. Belbachir and I. E. Bousbaa, Associated Lah numbers and $r$-Stirling numbers, Mathematics, (2014).

[6]

E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combin. Theory Ser. A, 15 (1973), 91-111.  doi: 10.1016/0097-3165(73)90038-1.

[7]

E. R. Canfield, Asymptotic normality in enumeration, Handbook of enumerative combinatories, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, (2015), 255–280.

[8]

C. A. Charalambides, Enumerative Combinatoric, CRC Press Series on Discrete Mathematics and its Applications, Chapman and Hall/CRC, Boca Raton, FL, 2002.

[9]

L. Comtet, Adanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.

[10]

D. Galvin, Asymptotic normality of some graph sequences, Graphs Combin., 32 (2016), 639-647.  doi: 10.1007/s00373-015-1596-4.

[11]

C. D. Godsil, Matching behavior is asymptotically normal, Combinatorica, 1 (1981), 369-376. 

[12]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics-A Foundation for Computer Science, 2$^{nd}$ editio, Addison-Wesley Publishing Company, Reading, MA, 1994.

[13]

L. H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Stattist., 38 (1967), 410-414.  doi: 10.1214/aoms/1177698956.

[14]

J. Kahn, A normal law for matchings, Combinatorica, 20 (2000), 339-391.  doi: 10.1007/PL00009835.

[15]

J. L. LebowitzB. PittelD. Ruelle and E. R. Speer, Central limit theorems, Lee-Yang zeros, and graph-counting polynomials, J. Combin. Theory Ser. A, 141 (2016), 147-183.  doi: 10.1016/j.jcta.2016.02.009.

[16]

J. LindsayT. Mansour and M. Shattuck, A new combinatorial interpretation of a $q$-analogue of the Lah numbers, J. Comb., 2 (2011), 245-264.  doi: 10.4310/JOC.2011.v2.n2.a4.

[17]

L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Apple. Math., 38 (2007), 542-560.  doi: 10.1016/j.aam.2006.02.003.

[18]

T. S. Motzkin, Sorting numbers for cylinders other classification numbers, Proc. Symp. Pure Math, 19 (1971), 167-176. 

[19]

S. B. Nandi and S. K. Dutta, On associated and generalized Lah numbers and applications to discrete distributions, Fibonacci Quart., 25 (1987), 128-136. 

[20]

K. Nowick, Asymptotic normality of graph statistics, J. Statist. Plann. Inference, 21 (1989), 209-222.  doi: 10.1016/0378-3758(89)90005-0.

[21]

M. Petkov$\check{s}$ek and T. Pisanski, Combinatorial interpretation of unsigned Stirling and Lah numbers, preprint, Univ. of Ljubljana.(Available on the Internet), 40 (2002).

[22]

A. Ruciński, The behaviour of $\frac{\binom{n}{k, ...k, n-ik}}{i!}$ is asymptotically normal, Discrete Math., 49 (1984), 287-290.  doi: 10.1016/0012-365X(84)90165-1.

[23]

C. Wagner, Generalized Stirling and Lah numbers, Discrete Math., 160 (1996), 199-218.  doi: 10.1016/0012-365X(95)00112-A.

[24]

Y. WangH.-X. Zhang and B.-X. Zhu, Asymptotic normality of Laplacian coefficients of graphs, J. Math. Anal. Appl., 455 (2017), 2030-2037.  doi: 10.1016/j.jmaa.2017.06.052.

show all references

References:
[1]

J. C. Ahuja, Distributions of the Sum of Independent Decapitated Negative Binomial Variables, Ann. Math. Statist., 42 (1971), 383-384.  doi: 10.1214/aoms/1177693527.

[2]

J. C. Ahuja and E. A. Enneking, Concavity property and a recurrence relation for associated Lah nubers, Fibonacci Quart., 17 (1979), 158-161. 

[3]

P. Baldi and Y. Rinott, Asymptotic normality of some graph-related statistics, J. Appl. Probab., 26 (1989), 171-175.  doi: 10.2307/3214327.

[4]

P. Barry, Some observations on the Lah and Laguerre transforms of integer sequences, J. Integer Seq., 10 (2007), 18pp.

[5]

H. Belbachir and I. E. Bousbaa, Associated Lah numbers and $r$-Stirling numbers, Mathematics, (2014).

[6]

E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combin. Theory Ser. A, 15 (1973), 91-111.  doi: 10.1016/0097-3165(73)90038-1.

[7]

E. R. Canfield, Asymptotic normality in enumeration, Handbook of enumerative combinatories, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, (2015), 255–280.

[8]

C. A. Charalambides, Enumerative Combinatoric, CRC Press Series on Discrete Mathematics and its Applications, Chapman and Hall/CRC, Boca Raton, FL, 2002.

[9]

L. Comtet, Adanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.

[10]

D. Galvin, Asymptotic normality of some graph sequences, Graphs Combin., 32 (2016), 639-647.  doi: 10.1007/s00373-015-1596-4.

[11]

C. D. Godsil, Matching behavior is asymptotically normal, Combinatorica, 1 (1981), 369-376. 

[12]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics-A Foundation for Computer Science, 2$^{nd}$ editio, Addison-Wesley Publishing Company, Reading, MA, 1994.

[13]

L. H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Stattist., 38 (1967), 410-414.  doi: 10.1214/aoms/1177698956.

[14]

J. Kahn, A normal law for matchings, Combinatorica, 20 (2000), 339-391.  doi: 10.1007/PL00009835.

[15]

J. L. LebowitzB. PittelD. Ruelle and E. R. Speer, Central limit theorems, Lee-Yang zeros, and graph-counting polynomials, J. Combin. Theory Ser. A, 141 (2016), 147-183.  doi: 10.1016/j.jcta.2016.02.009.

[16]

J. LindsayT. Mansour and M. Shattuck, A new combinatorial interpretation of a $q$-analogue of the Lah numbers, J. Comb., 2 (2011), 245-264.  doi: 10.4310/JOC.2011.v2.n2.a4.

[17]

L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Apple. Math., 38 (2007), 542-560.  doi: 10.1016/j.aam.2006.02.003.

[18]

T. S. Motzkin, Sorting numbers for cylinders other classification numbers, Proc. Symp. Pure Math, 19 (1971), 167-176. 

[19]

S. B. Nandi and S. K. Dutta, On associated and generalized Lah numbers and applications to discrete distributions, Fibonacci Quart., 25 (1987), 128-136. 

[20]

K. Nowick, Asymptotic normality of graph statistics, J. Statist. Plann. Inference, 21 (1989), 209-222.  doi: 10.1016/0378-3758(89)90005-0.

[21]

M. Petkov$\check{s}$ek and T. Pisanski, Combinatorial interpretation of unsigned Stirling and Lah numbers, preprint, Univ. of Ljubljana.(Available on the Internet), 40 (2002).

[22]

A. Ruciński, The behaviour of $\frac{\binom{n}{k, ...k, n-ik}}{i!}$ is asymptotically normal, Discrete Math., 49 (1984), 287-290.  doi: 10.1016/0012-365X(84)90165-1.

[23]

C. Wagner, Generalized Stirling and Lah numbers, Discrete Math., 160 (1996), 199-218.  doi: 10.1016/0012-365X(95)00112-A.

[24]

Y. WangH.-X. Zhang and B.-X. Zhu, Asymptotic normality of Laplacian coefficients of graphs, J. Math. Anal. Appl., 455 (2017), 2030-2037.  doi: 10.1016/j.jmaa.2017.06.052.

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