| 冯红,李玉双.一个高斯系数恒等式的组合证明[J].,2008,(1):154-156 |
| 一个高斯系数恒等式的组合证明 |
| Combinatorial proof for a Gaussian coefficient identity |
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| DOI：10.7511/dllgxb200801028 |
| 中文关键词: 第二类广义二项式系数 高斯系数 二项式系数 |
| 英文关键词: generalized binomial coefficients of the second kind Gaussian coefficient binomial coefficient °C |
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| 中文摘要: |
| 高斯系数恒等式的传统证明方法包括代数证明和子集-子空间模拟. 把高斯系数看做Konvalina定义的重量为W=(w 1〓w 2〓…〓w n)(w i=q i)的第二类广义二项式系数，结合对偶选择，即从集合{1,2,…,n-k+1}中可重复地选取k个盒子与从{1,2,…,k+1}中可重复地选取n-k个盒子一一对应，通过证明一种选择与它的对偶选择具有相同的重量，从而给出一个高斯系数恒等式的组合证明. 由0，1，0，1组成的选择序列表示对于等式的证明起到了至关重要的作用. 当q=1时得到对应的普通二项式系数恒等式. 这种证明方法深刻地揭示了高斯系数和二项式系数之间的组合联系. |
| 英文摘要: |
| The traditional proofs for Gaussian coefficient identities are often algebraic or mimic subset proofs. Interpreting Gaussian coefficients as the generalized binomial coefficients of the second kind with weight W=(w 1〓w 2〓…〓w n)(w i=q i) defined by Konvalina, combining dual selection, which is a selection of k boxes with repetition from the set {1,2,…, n-k+1 }, can be uniquely associated with an n-k selection with repetition from the set {1,2,…, k+1 }, and the results show that a selection and its dual have the same weight to give a combinatorial proof for a Gaussian coefficient identity. The sequence representation of a selection consisting of 0, 1, 0, 1, plays an important role in the proof. When q=1 , this Gaussian coefficient identity changes the corresponding binomial coefficient identity. This method reveals the combinatorial connections between the binomial coefficients and the Gaussian coefficients. |
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