论文标题
整数序列和单一理想
Integer Sequences and Monomial Ideals
论文作者
论文摘要
令$ \ mathfrak {s} _n $为$ [n] = \ {1，\ ldots，n \} $的所有排列的集合，让$ w $是由\ athfrak in \ mathfrak {s}} _n $避免132和312-Patterns组成的子集$σ\ in \ mathfrak in \ mathfrak in \ mathfrak in \ mathfrak in。单一的理想$ i_w = \ left \ langle \ mathbf {x}^σ= \ prod_ {i = 1}^n x_i^{σ（i）}：σ\ in w \ right \ rangle \ rangle $ in polynomial ring unge $ r = k [x_1 infort $ r = k [x_1 infort a fiffe a fiffe a in fiffe a in fifts a $ a $ y hyy y hyper a y hyroys a y hyy x $ a y hyy x__n] （某些多电体辅助理想的变体，Proc。IndianAcad。Sci。（MathSci。Vol。126，No. 4，（2016），479-500）。亚历山大Dual dual $ i_w^{[\ Mathbf {n}} $ of $ i_w $ of $ i_W $ nminim nminim inim nimim Inim在第一个Barycentric $ \ Mathbf {Bd}（δ__{N-1}）$的分辨率上支持$ N-1 $ -SIMPLEX $δ_{n-1 {n-1} $的分辨率。顶点的植物标记的单峰森林套装$ [n] $。 \ [\ dim_k \ left（\ frac {r} {i_w^^{[\ mathbf {n}]}}}}} \ right）= \ sum_ {r = 1}^n r！ n}）$是矩阵$ [m_ {ij}] $的永久性，$ m_ {ii} = i $和$ m_ {ij} = 1 $ for $ i \ ne j $。对于各种子集$ s $ s $ of $ \ mathfrak {s} _n $由避免模式组成的排列，相应的整数序列$ \ left \ left \ lbrace \ dim_k \ left（\ frac {rac {r} {r} {i_s^{i_s^{ \ right \ rbrace_ {n = 1}^{\ infty} $被标识。
Let $\mathfrak{S}_n$ be the set of all permutations of $[n]=\{1,\ldots,n\}$ and let $W$ be the subset consisting of permutations $σ\in \mathfrak{S}_n$ avoiding 132 and 312-patterns. The monomial ideal $I_W = \left\langle \mathbf{x}^σ = \prod_{i=1}^n x_i^{σ(i)} : σ\in W \right\rangle $ in the polynomial ring $R = k[x_1,\ldots,x_n]$ over a field $k$ is called a hypercubic ideal in the article (Certain variants of multipermutohedron ideals, Proc. Indian Acad. Sci.(Math Sci. Vol. 126, No.4, (2016), 479-500). The Alexander dual $I_W^{[\mathbf{n}]}$ of $I_W$ with respect to $\mathbf{n}=(n,\ldots,n)$ has the minimal cellular resolution supported on the first barycentric subdivision $\mathbf{Bd}(Δ_{n-1})$ of an $n-1$-simplex $Δ_{n-1}$. We show that the number of standard monomials of the Artinian quotient $\frac{R}{I_W^{[\mathbf{n}]}}$ equals the number of rooted-labelled unimodal forests on the vertex set $[n]$. In other words, \[ \dim_k\left(\frac{R}{I_W^{[\mathbf{n}]}}\right) = \sum_{r=1}^n r!~s(n,r) = {\rm Per}\left([m_{ij}]_{n \times n} \right),\] where $s(n,r)$ is the (signless) Stirling number of the first kind and ${\rm Per}([m_{ij}]_{n \times n})$ is the permanent of the matrix $[m_{ij}]$ with $m_{ii}=i$ and $m_{ij}=1$ for $i \ne j$. For various subsets $S$ of $\mathfrak{S}_n$ consisting of permutations avoiding patterns, the corresponding integer sequences $\left\lbrace \dim_k\left(\frac{R}{I_S^{[\mathbf{n}]}}\right) \right\rbrace_{n=1}^{\infty}$ are identified.
