A COMBINATORIAL BIJECTION ON k-NONCROSSING PARTITIONS

For any integer k >= 2, we prove combinatorially the following Euler (binomial) transformation identity NCn+1(k) (t) = t Sigma(n)(i=0) (n(i)) NWi(k) (t), where NCm(k) (t) (resp. NWm(k) (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.