Geodesic
On the diagonal ideal of $(\mathbb{C}^2)^n$ and $q,t$-Catalan numbers
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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Let $I_n$ be the (big) diagonal ideal of $(\mathbb{C}^2)^n$. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ spanned by a minimal set of generators for $I_n$. We give simple upper bounds on $\textrm{dim} (M_n)_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\textrm{dim} (M_n)_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $(M_n)_{d_1, d_2}$. 
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     author = {Lee, Kyungyong and Li, Li},
     title = {On the diagonal ideal of $(\mathbb{C}^2)^n$ and $q,t${-Catalan} numbers},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2838},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2838/}
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Lee, Kyungyong; Li, Li. On the diagonal ideal of $(\mathbb{C}^2)^n$ and $q,t$-Catalan numbers. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2838. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2838/

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