Blocking Mixed-Level Factorials withSAS

λ-Factorials of n

where Dk is the number of permutations with no fixed points on {1, 2, . . . , k}. In the paper, we utilize the λ-factorials of n, defined by Eriksen, Freij and Wästlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and two algebraic proofs. Using the umbral representation of our generalized identity and Abel’s...

Factorials as sums

Products of Three Factorials

We study a question of Erdős and Graham on products of three factorials being a square.

Logarithmic Tree Factorials

To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargava’s factorials to a natural combinatorial setting suitable for studying questions around generalized factorials. We discuss several basic aspects of the framework in this paper. In particular, we relate the growth of the sequence of logarithmic fac...

Uniformity in Fractional Factorials