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Stirling permutations containing a single pattern of length three. (English) Zbl 1419.05009
Summary: We derive explicit formulæ for the number of \(k\)-Stirling permutations containing a single occurrence of a single pattern of length three as well as expressions for the corresponding generating functions. Furthermore, asymptotic results for these numbers are given.
MSC:
05A05 Permutations, words, matrices
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[1] M. Albert, R. Aldred, M. Atkinson, C. Handley and D. Holton, Permutations of a multiset avoiding permutations of length 3, European J. Combin. 22 (2001), 1021-1031. · Zbl 0988.05005
[2] C. Banderier, M. Bousquet-M´elou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Math. 246 (2002), 29-55. M. KUBA AND A. PANHOLZER / AUSTRALAS. J. COMBIN. 74 (2) (2019), 216-239238 · Zbl 0997.05007
[3] J. Barbero, J. Salas and E. Villase˜nor, Generalized Stirling permutations and forests: higher-order Eulerian and Ward numbers, Electr. J. Combin. 22(3) (2015), #P3.37.
[4] M. Bona, The number of permutations with exactly r 132-subsequences is P recursive in the size!, Adv. Appl. Math. 18 (1997), 510-522. · Zbl 0879.05006
[5] A. Burstein, A short proof for the number of permutations containing pattern 321 exactly once, Electr. J. Combin. 18(2) (2011), #P21. · Zbl 1229.05006
[6] P. Br¨and´en and T. Mansour, Finite automata and pattern avoidance in words, J. Combin. Theory Ser. A 110(1) (2005), 127-145.
[7] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009. · Zbl 1165.05001
[8] I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory Ser. A 24(1) (1978), 24-33.
[9] S. Heubach and T. Mansour, Avoiding patterns of length three in compositions and multiset permutations, Adv. Appl. Math. 36 (2006), 156-174. · Zbl 1088.05010
[10] S. Janson, M. Kuba and A. Panholzer, Generalized Stirling permutations, families of increasing trees and urn models, J. Combin. Theory Ser. A 118 (2011), 94-114. · Zbl 1230.05100
[11] S. Kitaev, Patterns in Permutations and Words, Springer Verlag (EATCS monographs in Theoretical Computer Science book series), 2011. · Zbl 1257.68007
[12] M. Kuba and A. Panholzer, Enumeration formulæ for pattern restricted Stirling permutations, Discrete Math. 312 (2012), 3179-3194. · Zbl 1252.05011
[13] M. Kuba and A. Panholzer, Analysis of statistics for generalized Stirling permutations, Combin. Probab. Comput. 20 (2011), 875-910. · Zbl 1233.05014
[14] S.-M. Ma and T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, Discrete Math. 338 (2015), 1468-1472. · Zbl 1310.05009
[15] T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, Adv. Appl. Math. 28 (2002), 185-195. · Zbl 1005.05001
[16] T. Mansour and A. Vainshtein, Restricted permutations and Chebyshev polynomials, S´eminaire Lotharingien de Combinatoire 47, Article B47c, 2002.
[17] J. Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math. 152(1-3) (1996), 307-313. · Zbl 0852.05009
[18] J. Noonan and D. Zeilberger, The enumeration of permutations with a prescribed number of “forbidden” patterns, Adv. Appl. Math. 17(4) (1996), 381- 407. M. KUBA AND A. PANHOLZER / AUSTRALAS. J. COMBIN. 74 (2) (2019), 216-239239 · Zbl 0974.05001
[19] S. K. Park, The r-multipermutations, J. Combin.Theory Ser. A 67(1) (1994), 44-71. · Zbl 0804.05001
[20] H. Prodinger, The kernel method:a collection of examples, S´eminaire Lotharingien de Combinatoire 50, Article B50f, 2004.
[21] A. Robertson, Restricted Permutations from Catalan to Fine and Back, S´eminaire Lotharingien de Combinatoire 50, Article B50g, 2004.
[22] C. D. Savage and H. S. Wilf, Pattern avoidance in compositions and multiset permutations, Adv. Appl. Math. 36 (2) (2006), 194-201. · Zbl 1087.05002
[23] R. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999. · Zbl 0928.05001
[24] M.-J. Yang and D.
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