A central limit theorem for a new statistic on permutations. (English) Zbl 1390.60082
Summary: This paper does three things: It proves a central limit theorem for novel permutation statistics (for example, the number of descents plus the number of descents in the inverse). It provides a clear illustration of a new approach to proving central limit theorems more generally. It gives us an opportunity to acknowledge the work of our teacher and friend B. V. Rao.
References:
| [1] | E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combinatorial Theory Ser. A, 15 (1973), 91-111. · Zbl 0242.05006 · doi:10.1016/0097-3165(73)90038-1 |
| [2] | A. Borodin, P. Diaconis and J. Fulman, On adding a list of numbers (and other one-dependent determinantal processes), Bull. Amer. Math. Soc. (N.S.), 47(4) (2010), 639-670. · Zbl 1230.05292 · doi:10.1090/S0273-0979-2010-01306-9 |
| [3] | S. Chatterjee, A new method of normal approximation. Ann. Probab., 36(4) (2008), 1584-1610. · Zbl 1159.62009 · doi:10.1214/07-AOP370 |
| [4] | S. Chatterjee and K. Soundararajan, Random multiplicative functions in short intervals, Int. Math. Res. Not., 2012(3) (2012), 479-492. · Zbl 1248.11056 · doi:10.1093/imrn/rnr023 |
| [5] | L. Carlitz, Eulerian numbers and polynomials, Math. Mag., 32 (1958), 247-260. · Zbl 0092.06601 · doi:10.2307/3029225 |
| [6] | L Carlitz, D. P. Roselle and R. A. Scoville, Permutations and sequences with repetitions by number of increases, J. Combinatorial Theory, 1 (1966), 350-374. · Zbl 0304.05002 · doi:10.1016/S0021-9800(66)80057-1 |
| [7] | L. H. Y. Chen and Q.-M. Shao, Normal approximation under local dependence, Ann. Probab., 32(3A) (2004), 1985-2028. · Zbl 1048.60020 · doi:10.1214/009117904000000450 |
| [8] | M. A. Conger, A refinement of the Eulerian numbers, and the joint distribution of π(1) and Des(π) in Sn. Ars Combin., 95 (2010), 445-472. · Zbl 1249.05001 |
| [9] | D. E. Critchlow, Metric methods for analyzing partially ranked data, Springer Science & Business Media, (2012). · Zbl 0589.62041 |
| [10] | F. N. David and D. E. Barton, Combinatorial chance, Hafner Publishing Co., New York, (1962). |
| [11] | P. Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics, Hayward, CA, (1988). · Zbl 0695.60012 |
| [12] | J. Fulman, Descent identities, Hessenberg varieties, and the Weil conjectures, J. Combin. Theory Ser. A, 87(2) (1999), 390-397. · Zbl 0948.05007 · doi:10.1006/jcta.1999.2964 |
| [13] | Fulman, J., Stein’s method and non-reversible Markov chains, 69-77 (2004) |
| [14] | A. M. Garsia and I. Gessel, Permutation statistics and partitions. Adv. in Math., 31(3) (1979), 288-305. · Zbl 0431.05007 · doi:10.1016/0001-8708(79)90046-X |
| [15] | R. L. Graham, D. E. Knuth and O. Patashnik, Concrete mathematics. A foundation for computer science, Second edition, Addison-Wesley Publishing Company, Reading, MA, (1994). · Zbl 0836.00001 |
| [16] | L. H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Statist., 38 (1967), 410-414. · Zbl 0154.43703 · doi:10.1214/aoms/1177698956 |
| [17] | J. I. Marden, Analyzing and modeling rank data, CRC Press, (1996). · Zbl 0853.62006 |
| [18] | T. K. Petersen, Two-sided Eulerian numbers via balls in boxes. Math. Mag., 86(3) (2013), 159-176. · Zbl 1293.05004 · doi:10.4169/math.mag.86.3.159 |
| [19] | T. K. Petersen, Eulerian numbers, Birkhäuser/Springer, New York, (2015). · Zbl 1337.05001 · doi:10.1007/978-1-4939-3091-3 |
| [20] | J. Pitman, Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Combin. Theory Ser. A, 77(2) (1997), 279-303. · Zbl 0866.60016 · doi:10.1006/jcta.1997.2747 |
| [21] | D. Rawlings, Enumeration of permutations by descents, idescents, imajor index, and basic components. J. Combin. Theory Ser. A, 36(1) (1984), 1-14. · Zbl 0527.05005 · doi:10.1016/0097-3165(84)90074-8 |
| [22] | N. J. A. Sloane, A handbook of integer sequences, Academic Press, New York-London, (1973). · Zbl 0286.10001 |
| [23] | Stanley, R.; Aigner, M. (ed.), Eulerian partitions of a unit hypercube, 49 (1977), Dordrecht/Boston · Zbl 0359.05001 |
| [24] | R. P. Stanley, Enumerative combinatorics, Volume 1, Second edition, Cambridge University Press, Cambridge, (2012). · Zbl 1247.05003 |
| [25] | S. M. Stigler, Estimating serial correlation by visual inspection of diagnostic plots, Amer. Statist., 40(2) (1986), 111-116. · Zbl 0603.62094 |
| [26] | V. A. Vatutin, Limit theorems for the number of ascending segments in random permutations generated by sorting algorithms. Discrete Math. Appl., 4(1) (1994), 31-44. · Zbl 0801.60003 · doi:10.1515/dma.1994.4.1.31 |
| [27] | V. A. Vatutin, The numbers of ascending segments in a random permutation and in one inverse to it are asymptotically independent, Discrete Math. Appl., 6(1) (1996), 41-52. · Zbl 0847.05002 · doi:10.1515/dma.1996.6.1.41 |
| [28] | V. A. Vatutin and V. G. Mikhailov, On the number of readings of random nonequiprobable files under stable sorting, Discrete Math. Appl., 6(3) (1996), 207-223. · Zbl 0868.68039 · doi:10.1515/dma.1996.6.3.207 |
| [29] | D. Warren and E. Seneta, Peaks and Eulerian numbers in a random sequence. J. Appl. Probab., 33(1) (1996), 101-114. · Zbl 0845.60035 · doi:10.1017/S0021900200103754 |
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