Cyclic compositions and cycles of the hypercube

Abstract

The partition graph of a positive integer n, \(P_n\), is the graph whose vertices are the cyclic compositions of n and two vertices are adjacent if one composition is obtained from the other one by replacing two cyclically consecutive parts by their sum. In this paper we introduce and investigate the notions of singular cyclic composition and singular edge of \(P_n\). We associate with every singular edge and every cycle of \(P_n\), whose vertices are aperiodic cyclic compositions of n, a cycle or a set of disjoint cycles of equal length of the hypercube \(Q_n\).

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Chao, C.Y.: On a type of circulants. Linear Algebra Appl. 6, 241–248 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Chen, W.Y.C., Louck, J.D.: Necklaces, MSS sequences and DNA sequences. Adv. Appl. Math. 18, 18–32 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ferrari, M.M., Zagaglia Salvi, N.: Aperiodic compositions and classical integer sequences. J. Integer Seq. 20, 3 (2017)

  4. 4.

    Jiang, Z., Savage, C.D.: On the existence of symmetric chain decomposition in a quotient of the Boolean lattice. Discrete Math. 309, 5278–5283 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Knopfmacher, A., Robbins, N.: Some properties of cyclic compositions. Fibonacci Quart. 48(3), 249–255 (2010)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Knopfmacher, A., Luca, F., Robbins, N.: On cyclic compositions of positive integers. Aequat. Math. 82(1), 111–122 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Kramer, A.V.: A particular Hamiltonian cycle on middle levels in the de Bruijn digraph. Discrete Math. 312, 608–613 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Riordan, J.: An Introduction to Combinatorial Analysis. Wiley, New York (1958)

    Google Scholar 

  9. 9.

    Sommerville, D.Y.M.: On certain periodic properties of cyclic compositons of integers. Proc. Lond. Math. Soc. 263–313 (1909)

  10. 10.

    Wang, T.M., Savage, C.D.: A Gray code for necklaces of fixed density. SIAM J. Discrete Math. 9(4), 654–673 (1996)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to N. Zagaglia Salvi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ferrari, M.M., Salvi, N.Z. Cyclic compositions and cycles of the hypercube. Aequat. Math. 92, 671–682 (2018). https://doi.org/10.1007/s00010-018-0554-7

Download citation

Keywords

  • Composition
  • Cyclic composition
  • Singular composition
  • Singular cyclic composition
  • Partition graph
  • Hypercube
  • Necklace
  • Quotient of the Boolean lattice

Mathematics Subject Classification

  • Primary 05A17
  • Secondary 05C38