Mansour, T., Shattuck, M. (2018). Combinatorial parameters on bargraphs of permutations. Transactions on Combinatorics, 7(2), 1-16. doi: 10.22108/toc.2017.102359.1483

Toufik Mansour; Mark Shattuck. "Combinatorial parameters on bargraphs of permutations". Transactions on Combinatorics, 7, 2, 2018, 1-16. doi: 10.22108/toc.2017.102359.1483

Mansour, T., Shattuck, M. (2018). 'Combinatorial parameters on bargraphs of permutations', Transactions on Combinatorics, 7(2), pp. 1-16. doi: 10.22108/toc.2017.102359.1483

Mansour, T., Shattuck, M. Combinatorial parameters on bargraphs of permutations. Transactions on Combinatorics, 2018; 7(2): 1-16. doi: 10.22108/toc.2017.102359.1483

Combinatorial parameters on bargraphs of permutations

^{1}Department of Mathematics, University of Tennessee, Knoxville, TN, USA

^{2}Mathematics Department, University of Tennessee, Knoxville, TN, USA

Abstract

In this paper, we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which $\mathcal{S}_n$ is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases.

[1] A. Blecher, C. Brennan and A. Knopfmacher, Levels in bargraphs, Ars Math. Contemp., 9 (2015) 297–310.

[2] A. Blecher, C. Brennan and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. S. Afr., 71 (2016) 97–103.

[3] A. Blecher, C. Brennan and A. Knopfmacher, Combinatorial parameters in bargraphs, Quaest. Math., 39 (2016) 619–635.

[4] E. Deutsch and S. Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, Discrete Appl. Math., 221 (2017) 54–66.

[5] S. Fereti´ c, A perimeter enumeration of column-convex polyominoes, Discrete Math. Theor. Comput. Sci., 9 (2007) 57–83.

[6] A. Geraschenko, An investigation of skyline polynomials, http://people.brandeis.edu/ gessel/47a/geraschenko.pdf.

[7] S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Discrete Math. Appl., Chapman & Hall/CRC, Boca Raton, (2009).

[8] Q.-M. Luo, An explicit formula for the Euler numbers of higher order, Tamkang J. Math., 36 (2005) 315–317.

[9] T. Mansour, Enumeration of words by sum of differences between adjacent letters, Discrete Math. Theor. Comput. Sci., 11 (2009) 173–185.

[10] J. Osborn and T. Prellberg, Forcing adsorption of a tethered polymer by pulling, J. Stat. Mech., 2010 (2010) 1–18.

[11] T. Prellberg and R. Brak, Critical exponents from nonlinear functional equations for partially directed cluster models, J. Stat. Phys., 78 (1995) 701–730.

[12] N. J. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2010.

[13] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001.