Integer Sequences: Irregular Arrays and Intra-Block Permutations

This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor diagonal method is proposed. We also define and describe intra-block permutations of natural numbers. Generalizations of reluctant sequences are introduced, namely generalized reluctant sequences and generalized reverse reluctant sequences. Explicit formulas are presented for these sequences. The article provides numerous examples to illustrate all statements.