Sequences Derived from The Symmetric Powers of $\{1,2,\ldots,k\}$

TL;DR

This paper investigates sequences derived from the symmetric powers of finite sets, providing recursive formulas for specific cases and identifying key initial values needed for sequence construction.

Contribution

It introduces a new sequence based on symmetric powers and derives recursive formulas for small values of k, highlighting minimal initial data needed.

Findings

Recursive formulas for sequences when k=2 to 8

Identification of key initial values for sequence construction

Explicit connection between symmetric powers and sequence generation

Abstract

For a fixed integer $k$, we define a sequence $A_k=(a_k(n))_{n\geq0}$ and a corresponding sparse subsequence $S_k$ using the cardinality of the $n$-th symmetric power of the set $\{1,2,\ldots, k\}$. For $k\in\{2,\dots,8\}$, we find recursive formulas for $S_k$, and show that the values $a_{k}(0)$, $a_{k}(1)$, and $a_{k}(3)$ are sufficient for constructing $A_{k}$.