A Combinatorial Interpretation of the Catalan and Bell Number Difference Tables

Combinatorial proofs of some Stirling number formulas

“…In the final section, we provide a further interpretation for the related alternating sum n i=0 (−1) n−i n i c i , where c i denotes the i-th Catalan number. In particular, we show that it can be thought of in terms of the inclusion-exclusion principle, as requested in [6].…”

“…A partition is said to be non-crossing (see [4]) if its canonical form contains no subsequences of the form abab where a < b (i.e., if it avoids all occurrences of the pattern 1212). By the definitions from [6], an NINS n-column is seen to be equivalent to a non-crossing partition of the same length since, when an n-column is non-skipping, it can be shown that the interlocking property is equivalent to the non-crossing property. Thus, Theorem 3.2 in [6] is equivalent to the following combinatorial interpretation of K n :…”

“…By the definitions from [6], an NINS n-column is seen to be equivalent to a non-crossing partition of the same length since, when an n-column is non-skipping, it can be shown that the interlocking property is equivalent to the non-crossing property. Thus, Theorem 3.2 in [6] is equivalent to the following combinatorial interpretation of K n :…”

“…In [6], it was requested to provide a direct combinatorial proof of Theorem 3.2. Using our interpretation for K n in terms of non-crossing partitions, one may provide such a proof as follows.…”

“…That K n counts all non-crossing partitions of length n having no two adjacent letters the same now follows from an application of the inclusion-exclusion principle. In fact, it is further seen that the sum j i=0 (−1) i j i c n−i counts all non-crossing partitions of length n in which no letter equals its successor among the first j positions in analogy to Theorem 2.3 in [6].…”