[Paper Review] Periodicity of the last digits of some combinatorial sequences

István Mező|arXiv (Cornell University)|Aug 7, 2013

Advanced Combinatorial Mathematics29 references31 citations

TL;DR

This paper establishes the periodicity of the last digits of various combinatorial sequences derived from restricted set partitions and permutations, including r-Fubini numbers, restricted Fubini numbers, restricted factorials, and associated Fubini numbers. It proves that these sequences exhibit modular periodicity modulo 10, with periods of 4, 5, or 20 depending on the sequence, extending earlier results on Fubini numbers and providing combinatorial and algebraic proofs for these congruences.

ABSTRACT

In 1962 O. A. Gross proved that the last digits of the Fubini numbers (or surjective numbers) have a simple periodicity property. We extend this result to a wider class of combinatorial numbers coming from restricted set partitions.

Motivation & Objective

• To extend O. A. Gross's 1962 result on the periodicity of Fubini numbers' last digits to a broader class of combinatorial sequences derived from restricted set partitions.
• To investigate whether similar periodic behavior in the last digits occurs for r-Fubini numbers, restricted Fubini numbers, restricted factorials, and associated Fubini numbers.
• To determine the exact period lengths and prove the modular congruences for these sequences using combinatorial and algebraic techniques.
• To explain why periodicity fails for certain sequences, such as r-Bell numbers and restricted Bell numbers with m > 3.
• To analyze the parity and modular properties of associated Fubini numbers, particularly proving they are always odd for n ≥ m.

Proposed method

• Define and analyze r-Fubini numbers as sums of r-Stirling numbers of the second kind multiplied by factorials, generalizing Fubini numbers.
• Introduce restricted Fubini numbers and restricted factorials by constraining block or cycle sizes in set partitions and permutations.
• Use combinatorial arguments and binomial coefficient identities to derive closed-form expressions for restricted Stirling numbers.
• Apply modular arithmetic, particularly modulo 10 and modulo 2, to analyze last digit behavior and prove periodicity.
• Prove periodicity via direct computation of differences Fn+T,r − Fn,r ≡ 0 (mod 10) and use induction and binomial sum manipulations.
• Establish parity results for associated Fubini numbers by analyzing the sum of k! times the number of partitions with k blocks of size at least m.

Experimental results

Research questions

• RQ1 Do r-Fubini numbers exhibit periodicity in their last digits, and if so, what is the period length?
• RQ2 Do restricted Fubini numbers and restricted factorials (with block or cycle size constraints) show periodicity in their last digits?
• RQ3 Why does periodicity in the last digits fail for r-Bell numbers and restricted Bell numbers when m > 3?
• RQ4 Are associated Fubini numbers (with minimum block size m) always odd for n ≥ m, and do they exhibit periodic behavior modulo 10?
• RQ5 What explains the longer period of 20 in associated Fubini numbers compared to the period 4 in standard Fubini numbers?

Key findings

• The last digits of r-Fubini numbers are periodic with period 4: Fn+4,r ≡ Fn,r (mod 10) for all n, r ≥ 1.
• For restricted Fubini numbers with block size at most m, Fn,≤m ≡ 0 (mod 10) for all n > 4 and m = 1, 2, 3, 4.
• Restricted factorials An,≤m (sums of restricted permutations) are periodic modulo 10 with period 5 for n > 2 and m = 2, 3, 4.
• Associated Fubini numbers Fn,≥m are always odd for n ≥ m, i.e., Fn,≥m ≡ 1 (mod 2).
• The last digits of associated Fubini numbers are periodic with period 20: Fn+20,≥m ≡ Fn,≥m (mod 10) for n ≥ 5 and m = 2, 3, 4, 5.
• The paper proves that for m > 4, restricted factorials An,≤m ≡ 0 (mod 10) for n > m, explaining the absence of periodicity in such cases.