Mathematics – Combinatorics
Scientific paper
2007-11-06
Discrete Mathematics 224 (2000) 15-35
Mathematics
Combinatorics
Scientific paper
The purpose of this paper is to find a new way to prove the $n!$ conjecture for particular partitions. The idea is to construct a monomial and explicit basis for the space $M_{\mu}$. We succeed completely for hook-shaped partitions, i.e., $\mu=(K+1,1^L)$. We are able to exhibit a basis and to verify that its cardinality is indeed $n!$, that it is linearly independent and that it spans $M_{\mu}$. We derive from this study an explicit and simple basis for $I_{\mu}$, the annihilator ideal of $\Delta_{\mu}$. This method is also successful for giving directly a basis for the homogeneous subspace of $M_{\mu}$ consisting of elements of 0 $x$-degree.
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