A bijection between certain non-crossing partitions and sequences

We present a bijection between non-crossing partitions of the set [2n+1][2n+1] into n+1n+1 blocks such that no block contains two consecutive integers, and the set of sequences View the MathML source{si}1n such that 1⩽si⩽i1⩽si⩽i, and if si=jsi=j, then si-r⩽j-rsi-r⩽j-r for 1⩽r⩽j-11⩽r⩽j-1.