Let's Expand Rota's Twelvefold Way For Counting Partitions!

TL;DR

This paper extends Rota's Twelvefold Way by creating a broader 6x5 table of partition counting scenarios, including new counts like Bell numbers and arrangements, with implications for combinatorial enumeration and sequence organization.

Contribution

It introduces a expanded 6x5 table of partition counting scenarios, adding new counts and relationships, and connects these to the OEIS sequences.

Findings

Expanded the original 12 scenarios to 30 counts.

Connected new counting sequences to OEIS entries.

Included teaching remarks and additional ordering considerations.

Abstract

Rota's Twelvefold Way gave formulas for the numbers of partitions which could be formed in twelve scenarios. This proposed AMM article expands Rota's 4 x 3 table. The resulting 6 x 5 table considers a broader collection of splitting-distributing-grouping-arranging scenarios, each of which can be visualized with the distribution of m items into certain kinds of bins. The additional counts or scenarios include: the Bell numbers B(m), the partition numbers p(m), arrangements of m books on b shelves, standings of m teams in a league, arrangements of m books into b scattered stacks, and pairings of 2m items. Teaching remarks are included. The two additional rows (due to K. Bogart) consider ordering the items within the bins. One additional column distributes the items into an unspecified number of bins, each receiving at least one item. The other (due to T. Brylawski) distributes the items into bins such that the number of bins containing a given number of items is specified. The quotient and summation relationships amongst the thirty counts are stated. A closely related table formed by the same six rows and seven certain columns is used to complete and to organize a 6 x 7 family of counting sequences in the On-Line Encyclopedia of Integer Sequences.