---
tag: PS7NS1AT2H
kind: concept
name: Telephone number (mathematics)
aliases: [involution-number]
sources:
  - wikipedia: en:35258497@r1370481196
    locator: Telephone number (mathematics)
    note: adopted
    mention: wikipedia://en/35258497
    sections: [Summary]
same_as:
  wikidata: Q7696507
  wikipedia: Telephone number (mathematics)
inventory: [WJ9BKNG2R8]
---

## Summary

In mathematics, the **telephone numbers** or the **involution numbers** form a sequence of integers that count the ways $n$ people can be connected by person-to-person telephone calls. These numbers also describe the number of [[matchings|0MJ04A50QW]] (the Hosoya index) of a [[complete graph|03WKB2H474]] on $n$ vertices, the number of [[permutations|05RWFKVTJ7]] on $n$ elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with $n$ cells, and the sum of the degrees of the irreducible representations of the [[symmetric group|0515A6D2Y1]]. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values (starting from $n = 0$)