Geodesic
A generalization of $(q,t)$-Catalan and nabla operators
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008).

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We introduce non-commutative analogs of $k$-Schur functions and prove that their images by the non-commutative nabla operator $\blacktriangledown$ is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual $(q,t)$-Catalan polynomials by computing the image of certain commutative $k$-Schur functions by the commutative nabla operator $\nabla$. In some particular cases, we give a combinatorial interpretation of these polynomials in terms of nested quantum Dick paths. 
@article{DMTCS_2008_special_255_a5,
     author = {Bergeron, N. and Descouens, F. and Zabrocki, M.},
     title = {A generalization of $(q,t)${-Catalan} and nabla operators},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)},
     year = {2008},
     doi = {10.46298/dmtcs.3597},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3597/}
}
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Bergeron, N.; Descouens, F.; Zabrocki, M. A generalization of $(q,t)$-Catalan and nabla operators. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008). doi : 10.46298/dmtcs.3597. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3597/

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