For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identity

$${\rm{NC}}_{n + 1}^{(k)}(t) = t\sum\limits_{i = 0}^n {\left({\matrix{n \cr i \cr}} \right)} {\rm{NW}}_i^{(k)}(t),$$

where NC ^(k)_m (t) (resp. NW ^(k)_m (t)) is the sum of weights, t^{number of blocks}, of partitions of {1,…,m} without k-crossings (resp. enhanced k-crossings). The special k = 2 and t = 1 case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for k = 3 and t = 1, arising naturally in a recent study of pattern avoidance in ascent sequences and inversion sequences, was proved only analytically.

中文翻译：

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k-非交叉分区的组合双射

对于任何整数k ≥ 2，我们组合证明以下欧拉（二项式）变换恒等式

$${\rm{NC}}_{n + 1}^{(k)}(t) = t\sum\limits_{i = 0}^n {\left({\matrix{n \cr i \ cr}} \right)} {\rm{NW}}_i^{(k)}(t),$$

其中 NC ^{( k )} _m ( t ) (resp. NW ^{( k )} _m ( t )) 是没有k交叉点的 {1,…,m} 的分区的权重和t^块数（分别增强k交叉口）。特殊的k = 2 和t = 1 情况，断言 Motzkin 数的欧拉变换是加泰罗尼亚数，是由 Donaghey 1977 发现的。k = 3 和t = 1 的结果，在最近对上升模式避免的研究中自然产生序列和反转序列，仅在分析上得到证明。