Pattern containment and avoidance have been extensively studied in permutations. Recently, analogous questions have been examined for colored permutations and circular permutations. In this note, we explore these problems in colored circular permutations. We present some interesting observations, some of which are direct generalizations of previously established results. We also raise some questions and propose directions for future study.
References
M. H. Albert, M. D. Atkinson, C. C. Handley, D. A. Holton, and W. Stromquist, “On packing densities of permutations”, Electron. J. Combin. 9:1 (2002), art. id. R5. MR1887086 0982.05005 M. H. Albert, M. D. Atkinson, C. C. Handley, D. A. Holton, and W. Stromquist, “On packing densities of permutations”, Electron. J. Combin. 9:1 (2002), art. id. R5. MR1887086 0982.05005
R. Arratia, “On the Stanley–Wilf conjecture for the number of permutations avoiding a given pattern”, Electron. J. Combin. 6 (1999), art. id. N1. MR1710623 0922.05002 R. Arratia, “On the Stanley–Wilf conjecture for the number of permutations avoiding a given pattern”, Electron. J. Combin. 6 (1999), art. id. N1. MR1710623 0922.05002
M. J. Bannister, W. E. Devanny, and D. Eppstein, “Small superpatterns for dominance drawing”, pp. 92–103 in 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics $($ANALCO$)$, edited by M. Drmota and M. D. Ward, SIAM, Philadelphia, PA, 2014. MR3248357 M. J. Bannister, W. E. Devanny, and D. Eppstein, “Small superpatterns for dominance drawing”, pp. 92–103 in 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics $($ANALCO$)$, edited by M. Drmota and M. D. Ward, SIAM, Philadelphia, PA, 2014. MR3248357
M. Bóna, Combinatorics of permutations, 2nd ed., CRC Press, Boca Raton, FL, 2012. MR2919720 1255.05001 M. Bóna, Combinatorics of permutations, 2nd ed., CRC Press, Boca Raton, FL, 2012. MR2919720 1255.05001
A. Burstein, P. Hästö, and T. Mansour, “Packing patterns into words”, Electron. J. Combin. 9:2 (2002/03), art. id. R20. MR2028289 1031.05007 A. Burstein, P. Hästö, and T. Mansour, “Packing patterns into words”, Electron. J. Combin. 9:2 (2002/03), art. id. R20. MR2028289 1031.05007
D. Callan, “Pattern avoidance in circular permutations”, preprint, 2002. math/0210014v1 D. Callan, “Pattern avoidance in circular permutations”, preprint, 2002. math/0210014v1
H. Eriksson, K. Eriksson, S. Linusson, and J. Wästlund, “Dense packing of patterns in a permutation”, Ann. Comb. 11:3-4 (2007), 459–470. MR2376116 1141.05004 10.1007/s00026-007-0329-7 H. Eriksson, K. Eriksson, S. Linusson, and J. Wästlund, “Dense packing of patterns in a permutation”, Ann. Comb. 11:3-4 (2007), 459–470. MR2376116 1141.05004 10.1007/s00026-007-0329-7
D. Gray, “Bounds on superpatterns containing all layered permutations”, Graphs Combin. 31:4 (2015), 941–952. MR3357666 1316.05001 10.1007/s00373-014-1429-x D. Gray, “Bounds on superpatterns containing all layered permutations”, Graphs Combin. 31:4 (2015), 941–952. MR3357666 1316.05001 10.1007/s00373-014-1429-x
D. Gray and H. Wang, “Note on superpatterns”, Involve 9:5 (2016), 797–804. MR3541980 1343.05007 10.2140/involve.2016.9.797 euclid.involve/1511371070 D. Gray and H. Wang, “Note on superpatterns”, Involve 9:5 (2016), 797–804. MR3541980 1343.05007 10.2140/involve.2016.9.797 euclid.involve/1511371070
M. Just and H. Wang, “Note on packing patterns in colored permutations”, Online J. Anal. Comb. 11 (2016), art. id. 4. MR3486799 1334.05002 M. Just and H. Wang, “Note on packing patterns in colored permutations”, Online J. Anal. Comb. 11 (2016), art. id. 4. MR3486799 1334.05002
M. L. Liendo, Preferential arrangement containment in strict superpatterns, master's thesis, East Tennessee State University, 2012, https://dc.etsu.edu/etd/1428/. https://dc.etsu.edu/etd/1428/ M. L. Liendo, Preferential arrangement containment in strict superpatterns, master's thesis, East Tennessee State University, 2012, https://dc.etsu.edu/etd/1428/. https://dc.etsu.edu/etd/1428/
T. Mansour, “Pattern avoidance in coloured permutations”, Sém. Lothar. Combin. 46 (2001), art. id. B46g. MR1877633 0986.05004 T. Mansour, “Pattern avoidance in coloured permutations”, Sém. Lothar. Combin. 46 (2001), art. id. B46g. MR1877633 0986.05004
A. Miller, “Asymptotic bounds for permutations containing many different patterns”, J. Combin. Theory Ser. A 116:1 (2009), 92–108. MR2469250 1177.05012 10.1016/j.jcta.2008.04.007 A. Miller, “Asymptotic bounds for permutations containing many different patterns”, J. Combin. Theory Ser. A 116:1 (2009), 92–108. MR2469250 1177.05012 10.1016/j.jcta.2008.04.007
A. L. Price, Packing densities of layered patterns, Ph.D. thesis, University of Pennsylvania, 1997, https://search.proquest.com/docview/304421853. MR2695616 https://search.proquest.com/docview/304421853 A. L. Price, Packing densities of layered patterns, Ph.D. thesis, University of Pennsylvania, 1997, https://search.proquest.com/docview/304421853. MR2695616 https://search.proquest.com/docview/304421853
W. Stromquist, “Packing layered posets into posets”, preprint, 1993, http://walterstromquist.com/papers/POSETS.DOC. http://walterstromquist.com/papers/POSETS.DOC W. Stromquist, “Packing layered posets into posets”, preprint, 1993, http://walterstromquist.com/papers/POSETS.DOC. http://walterstromquist.com/papers/POSETS.DOC
