WORDS 2017: Combinatorics on Words pp 190-202

# Combinatorics of Cyclic Shifts in Plactic, Hypoplactic, Sylvester, and Related Monoids

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)

## Abstract

The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. For certain monoids connected with combinatorics, such as the plactic monoid (the monoid of Young tableaux) and the sylvester monoid (the monoid of binary search trees), connected components consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper discusses new results on the diameters of connected components of the cyclic shift graphs of the finite-rank analogues of these monoids, showing that the maximum diameter of a connected component is dependent only on the rank. The proof techniques are explained in the case of the sylvester monoid.

### Keywords

Cyclic shift Plactic monoid Sylvester monoid Binary search tree Cocharge

### References

1. 1.
Cain, A.J., Malheiro, A.: Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, and related monoids (in preparation)Google Scholar
2. 2.
Cain, A.J., Malheiro, A.: Deciding conjugacy in sylvester monoids and other homogeneous monoids. Int. J. Algebra Comput. 25(5), 899–915 (2015). doi:10.1007/978-3-642-40579-2_11
3. 3.
Choffrut, C., Mercaş, R.: The lexicographic cross-section of the plactic monoid is regular. In: Karhumäki, J., Lepistö, A., Zamboni, L. (eds.) WORDS 2013. LNCS, vol. 8079, pp. 83–94. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40579-2_11
4. 4.
Giraudo, S.: Algebraic and combinatorial structures on pairs of twin binary trees. J. Algebra 360, 115–157 (2012). doi:10.1016/j.jalgebra.2012.03.020
5. 5.
Hivert, F., Novelli, J.C., Thibon, J.Y.: The algebra of binary search trees. Theoret. Comput. Sci. 339(1), 129–165 (2005). doi:10.1016/j.tcs.2005.01.012
6. 6.
Hivert, F., Novelli, J.C., Thibon, J.Y.: Commutative combinatorial Hopf algebras. J. Algebraic Combin. 28(1), 65–95 (2007). doi:10.1007/s10801-007-0077-0
7. 7.
Krob, D., Thibon, J.Y.: Noncommutative symmetric functions IV: quantum linear groups and hecke algebras at $$q=0$$. J. Algebraic Combin. 6(4), 339–376 (1997). doi:10.1023/A:1008673127310
8. 8.
Lascoux, A., Schützenberger, M.P.: Le monoïde plaxique. In: Noncommutative structures in algebra and geometric combinatorics, pp. 129–156. No. 109 in Quaderni de “La Ricerca Scientifica”, CNR, Rome (1981). http://igm.univ-mlv.fr/berstel/Mps/Travaux/A/1981-1PlaxiqueNaples.pdf
9. 9.
Lothaire, M.: Algebraic Combinatorics on Words. No. 90 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2002)
10. 10.
Novelli, J.C.: On the hypoplactic monoid. Discrete Math. 217(1–3), 315–336 (2000). doi:10.1016/S0012-365X(99)00270-8
11. 11.
Priez, J.B.: A lattice of combinatorial Hopf algebras: binary trees with multiplicities. In: Formal Power Series and Algebraic Combinatorics. The Association. Discrete Mathematics & Theoretical Computer Science, Nancy (2013). http://www.dmtcs.org/pdfpapers/dmAS0196.pdf