The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.
翻译：在 $k_ t$- Dyck 路径中, 双向下行路径数, 由 $ {( 1, k), ( 1, - 1) = $ = $ = $ = $ = 美元 上方的阶梯组成的 Dyck 路径的概略化, 这样可以研究路径在 $y =- t$ 上方的阶梯( 弱化 ) 。 结果通过产生函数的方式双向地被证明, 并导致几个有趣的身份以及与其他组合结构的链接 。 特别是, $k_ t$- Dyck 路径和在编码理论中使用的穿孔代码( 双向矩阵) 和穿孔模式( 双向矩阵) 。 奇怪的是, 在对普通 Dyck 路径的限制下, 这产生了对加泰罗尼亚 数字的新组合解释 。
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