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Yet another note on notation

Mircea Dan Rus

TL;DR

The paper introduces and motivates the notation $n^{\{m\}}$ for the number of surjections from an $m$-set to an $n$-set, and proposes the label 'subpowers' to reflect a power-like structure underlying these counts. It situates the construction within the history of exponential notation and factorial powers, and establishes key identities $n^{\{m\}} = \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} k^{m}$ and $n^{\{m\}} = n!\genfrac{\{}{0pt}{}{m}{n}$, linking to Stirling numbers of the second kind via binomial inversion. The work develops fundamental properties, including recurrence relations, a binomial-type expansion, and a rich network of connections to finite differences, polynomial identities, sums of powers, Bernoulli and Fubini numbers, and extensions to complex and negative exponents, underscoring the broad applicability of the proposed notation. A closing note invokes a Fermat-like conjecture for subpowers, highlighting the depth and novelty of the framework. Overall, the paper provides a coherent, notation-driven lens that unifies diverse combinatorial and analytic phenomena around surjection counts and their algebraic relatives.

Abstract

Back in 1755, Euler explored an interesting array of numbers that now frequently appears in polynomial identities, combinatorial problems, and finite calculus, among other places. These numbers share a strong connection with well-known number families, such as those of Stirling, Bernoulli, and Fubini. Despite their importance, they often go unnoticed because of the lack of a specific name and standard notation. This paper aims to address this oversight by proposing an appropriate name and notation, aligned with established mathematical conventions, and supported by (we hope) strong enough arguments to facilitate their acceptance from the mathematical community.

Yet another note on notation

TL;DR

The paper introduces and motivates the notation for the number of surjections from an -set to an -set, and proposes the label 'subpowers' to reflect a power-like structure underlying these counts. It situates the construction within the history of exponential notation and factorial powers, and establishes key identities and , linking to Stirling numbers of the second kind via binomial inversion. The work develops fundamental properties, including recurrence relations, a binomial-type expansion, and a rich network of connections to finite differences, polynomial identities, sums of powers, Bernoulli and Fubini numbers, and extensions to complex and negative exponents, underscoring the broad applicability of the proposed notation. A closing note invokes a Fermat-like conjecture for subpowers, highlighting the depth and novelty of the framework. Overall, the paper provides a coherent, notation-driven lens that unifies diverse combinatorial and analytic phenomena around surjection counts and their algebraic relatives.

Abstract

Back in 1755, Euler explored an interesting array of numbers that now frequently appears in polynomial identities, combinatorial problems, and finite calculus, among other places. These numbers share a strong connection with well-known number families, such as those of Stirling, Bernoulli, and Fubini. Despite their importance, they often go unnoticed because of the lack of a specific name and standard notation. This paper aims to address this oversight by proposing an appropriate name and notation, aligned with established mathematical conventions, and supported by (we hope) strong enough arguments to facilitate their acceptance from the mathematical community.
Paper Structure (15 sections, 2 theorems, 44 equations, 1 figure, 2 tables)

This paper contains 15 sections, 2 theorems, 44 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The context
  3. The notation
  4. The name
  5. The first impression
  6. The values
  7. The recurrence
  8. The binomial expansion
  9. The bigger picture. Making connections
  10. Finite differences
  11. Some polynomial identities and their ramifications
  12. Sums of powers and subpowers
  13. Down the rabbit hole
  14. Subpowers with negative integer exponents
  15. Instead of conclusions

Key Result

Proposition 3

Given two sequencesThe sequences may also be finite, but they must have the same number of terms.$(a_{n})_{n\geq0}$ and $(b_{n})_{n\geq0}$, the following statements are equivalent: (i)$\quad b_{n}=\sum_{k=0}^{n}\dbinom{n}{k}a_{k}$, for all $n$;(ii)$\quad a_{n}=\sum_{k=0}^{n}(-1)^{n-k}\dbinom{n}{k}b

Figures (1)

  • Figure 1: The graphs of the subpower functions of real argument $x$, with base $n$ from $1$ to $5$.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Proposition 3
  • Remark 4
  • Theorem 5