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n-color 1-2 compositions of positive integers

  • School of Mathematics and Statistics, Hexi University, Zhangye 734000, China

Funds:  Supported by the National Natural Science Foundation of China (11461020).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.12.005
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  • Author Bio:

    GUO Yuhong, female, born in 1970, master/Prof. Research field: combinatorial number theory. E-mail:gyh7001@163.com

  • Received Date: 21 March 2014
  • Accepted Date: 05 September 2014
  • Rev Recd Date: 05 September 2014
  • Publish Date: 30 December 2015
    Abstract

  • Abstract

    An n-color 1-2 composition is defined as an n-color composition with only parts of size 1 or 2 of positive integer. An n-color 1-2 palindromic composition is an n-color 1-2 composition in which the parts are ordered such that they are read the same forward and backwards. Here the generating function, explicit formulas and recurrence relations for n-color 1-2 compositions and n-color 1-2 palindromic compositions were obtained. In addition, a relation between the number of n-color 1-2 compositions of ν and the number of n-color 1-2 palindromic compositions of ν was given.

    Abstract

    An n-color 1-2 composition is defined as an n-color composition with only parts of size 1 or 2 of positive integer. An n-color 1-2 palindromic composition is an n-color 1-2 composition in which the parts are ordered such that they are read the same forward and backwards. Here the generating function, explicit formulas and recurrence relations for n-color 1-2 compositions and n-color 1-2 palindromic compositions were obtained. In addition, a relation between the number of n-color 1-2 compositions of ν and the number of n-color 1-2 palindromic compositions of ν was given.
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    • References(14)
  • [1]
    Alladi K, Hoggatt V E. Compositions with ones and twos[J]. Fibonacci Quart, 1975, 13(3): 233-239.
    [2]
    Chinn P, Heubach S. Integer sequence related to compositions without 2s[J]. J Integer Seq, 2003, 6(2): Article 03.2.3.
    [3]
    Knopfmacher A, Richmond B. Compositions with distinct parts[J]. Aequationes Math, 1995,49(12):86-97.
    [4]
    Shapcott C. Part-products of 1-free integer compositions[J]. Electron J Combin, 2011, 18(1): paper 235.
    [5]
    Agarwal A K. n-Colour compositions[J]. Indian J Pure Appl Math, 2000, 31(11): 1 421-1 427.
    [6]
    Agarwal A K. An analogue of Eulers identity and new combinatorial properties of n-colour compositions[J]. J Comput Appl Math, 2003, 160(1-2): 9-15.
    [7]
    Narang G, Agarwal A K. Lattice paths and n-colour compositions[J]. Discrete Math, 2008, 308(9): 1 732-1 740.
    [8]
    Guo Y H. Some n-color compositions[J]. Journal of Integer Sequence, 2012, 15: Article 12.1.2.
    [9]
    Narang G, Agarwal A K. n-Colour self-inverse compositions[J]. Proc Indian Acad Sci Math Sci, 2006, 116(3): 257-266.
    [10]
    Guo Y H. n-Colour even compositions[J]. Ars Combina, 2013, 109(2): 425-432.
    [11]
    Guo Y H. n-Colour even self-inverse compositions[J]. Proc Indian Acad Sci Math Sci, 2010, 120(1):27-33.
    [12]
    Shapcott C. C-color compositions and palindromes[J]. Fibonacci Quart, 2012, 50(4): 297-303.
    [13]
    Hoggatt V E, Bicknell M. Palindromic composition[J]. Fibonacci Quart, 1975, 13(4): 350-356.
    [14]
    MacMahon P A. Combinatory Analysis, Vol. Ⅰ and Ⅱ[M]. New York: AMS Chelsea Publishing, 2001.
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    [1]
    Alladi K, Hoggatt V E. Compositions with ones and twos[J]. Fibonacci Quart, 1975, 13(3): 233-239.
    [2]
    Chinn P, Heubach S. Integer sequence related to compositions without 2s[J]. J Integer Seq, 2003, 6(2): Article 03.2.3.
    [3]
    Knopfmacher A, Richmond B. Compositions with distinct parts[J]. Aequationes Math, 1995,49(12):86-97.
    [4]
    Shapcott C. Part-products of 1-free integer compositions[J]. Electron J Combin, 2011, 18(1): paper 235.
    [5]
    Agarwal A K. n-Colour compositions[J]. Indian J Pure Appl Math, 2000, 31(11): 1 421-1 427.
    [6]
    Agarwal A K. An analogue of Eulers identity and new combinatorial properties of n-colour compositions[J]. J Comput Appl Math, 2003, 160(1-2): 9-15.
    [7]
    Narang G, Agarwal A K. Lattice paths and n-colour compositions[J]. Discrete Math, 2008, 308(9): 1 732-1 740.
    [8]
    Guo Y H. Some n-color compositions[J]. Journal of Integer Sequence, 2012, 15: Article 12.1.2.
    [9]
    Narang G, Agarwal A K. n-Colour self-inverse compositions[J]. Proc Indian Acad Sci Math Sci, 2006, 116(3): 257-266.
    [10]
    Guo Y H. n-Colour even compositions[J]. Ars Combina, 2013, 109(2): 425-432.
    [11]
    Guo Y H. n-Colour even self-inverse compositions[J]. Proc Indian Acad Sci Math Sci, 2010, 120(1):27-33.
    [12]
    Shapcott C. C-color compositions and palindromes[J]. Fibonacci Quart, 2012, 50(4): 297-303.
    [13]
    Hoggatt V E, Bicknell M. Palindromic composition[J]. Fibonacci Quart, 1975, 13(4): 350-356.
    [14]
    MacMahon P A. Combinatory Analysis, Vol. Ⅰ and Ⅱ[M]. New York: AMS Chelsea Publishing, 2001.
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