A combinatorial bijection on $k$-noncrossing partitions

Abstract: For any integer $k\geq2$, we prove combinatorially the following Euler (binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is the sum of weights, $t^\text{number of blocks}$, of partitions of ${1,\ldots,m}$ 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.