Injectively $k$-colored rooted forests

We enumerate injectively k-colored rooted forests with a given number of vertices of each color and a given sequence of root colors. We obtain from this result some new multi-parameter distributions of Fuss-Catalan numbers. As an additional application we enumerate triangulations of regular convex polygons according to their proper 3-coloring type.

[1]  André Raspaud,et al.  Injective coloring of planar graphs , 2009, Discret. Appl. Math..

[2]  Kenneth H. Rosen,et al.  Catalan Numbers , 2002 .

[3]  Bruno Courcelle,et al.  The Monadic Second order Logic of Graphs VI: on Several Representations of Graphs By Relational Structures , 1994, Discret. Appl. Math..

[4]  Jean-Christophe Aval,et al.  Multivariate Fuss-Catalan numbers , 2007, Discret. Math..

[5]  V. Chvátal.  A combinatorial theorem in plane geometry , 1975 .

[6]  André Raspaud,et al.  Obstructions to Injective Oriented Colourings , 2011, Electron. Notes Discret. Math..

[7]  Sherry H. F. Yan,et al.  The Raney numbers and (s, s+1)-core partitions , 2017, Eur. J. Comb..

[9]  Emeric Deutsch,et al.  Ordered trees with prescribed root degrees, node degrees, and branch lengths , 2004, Discret. Math..

[10]  John Riordan,et al.  Enumeration of Plane Trees by Branches and Endpoints , 1975, J. Comb. Theory, Ser. A.

[11]  E. Duchi,et al.  Fighting Fish: enumerative properties , 2016, 1611.04625.

[12]  Jan Kratochvíl,et al.  On the injective chromatic number of graphs , 2002, Discret. Math..

[13]  N. K. Sudev,et al.  A study on the injective coloring parameters of certain graphs , 2016, Discret. Math. Algorithms Appl..

[14]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[15]  André Raspaud,et al.  Good and Semi-Strong Colorings of Oriented Planar Graphs , 1994, Inf. Process. Lett..

[16]  P. Hilton,et al.  Catalan Numbers, Their Generalization, and Their Uses , 1991 .

[17]  Florence March,et al.  2016 , 2016, Affair of the Heart.

[18]  André Raspaud,et al.  Injective Oriented Colourings , 2009, WG.

[19]  Steve Fisk,et al.  A short proof of Chvátal's Watchman Theorem , 1978, J. Comb. Theory, Ser. B.