1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence A000085 in the
John Riordan provides the following explanation for these numbers: suppose that n people subscribe to a telephone service that can connect any two of them by a call, but cannot make a single call connecting more than two people. How many different patterns of connection are possible? For instance, with three subscribers, there are three ways of forming a single telephone call, and one additional pattern in which no calls are being made, for a total of four patterns. For this reason, the numbers counting how many patterns are possible are sometimes called the telephone numbers.
Every pattern of pairwise connections between n people defines an
permutation of the people that is its own inverse. In this permutation, each two people who call each other are swapped, and the people not involved in calls remain fixed in place. Conversely, every possible involution has the form of a set of pairwise swaps of this type. Therefore, the telephone numbers also count involutions. The problem of counting involutions was the original
combinatorial enumeration problem studied by Rothe in 1800 and these numbers have also been called involution numbers.
graph theory, a subset of the edges of a graph that touches each vertex at most once is called a
matching. Counting the matchings of a given graph is important in
chemical graph theory, where the graphs model molecules and the number of matchings is the
Hosoya index. The largest possible Hosoya index of an n-vertex graph is given by the
complete graphs, for which any pattern of pairwise connections is possible; thus, the Hosoya index of a complete graph on n vertices is the same as the nth telephone number.
A standard Young tableau
Ferrers diagram is a geometric shape formed by a collection of n squares in the plane, grouped into a
polyomino with a horizontal top edge, a vertical left edge, and a single monotonic chain of edges from top right to bottom left. A standard
Young tableau is formed by placing the numbers from 1 to n into these squares in such a way that the numbers increase from left to right and from top to bottom throughout the tableau.
According to the
Robinson–Schensted correspondence, permutations correspond one-for-one with ordered pairs of standard
Young tableaux. Inverting a permutation corresponds to swapping the two tableaux, and so the self-inverse permutations correspond to single tableaux, paired with themselves. Thus, the telephone numbers also count the number of Young tableaux with n squares. In
representation theory, the Ferrers diagrams correspond to the
irreducible representations of the
symmetric group of permutations, and the Young tableaux with a given shape form a basis of the irreducible representation with that shape. Therefore, the telephone numbers give the sum of the degrees of the irreducible representations.
A diagonally symmetric non-attacking placement of eight rooks on a chessboard
mathematics of chess, the telephone numbers count the number of ways to place n rooks on an n × nchessboard in such a way that no two rooks attack each other (the so-called
eight rooks puzzle), and in such a way that the configuration of the rooks is symmetric under a diagonal reflection of the board. Via the
Pólya enumeration theorem, these numbers form one of the key components of a formula for the overall number of "essentially different" configurations of n mutually non-attacking rooks, where two configurations are counted as essentially different if there is no symmetry of the board that takes one into the other.
first published in 1800 by
Heinrich August Rothe, by which they may easily be calculated.
One way to explain this recurrence is to partition the T(n) connection patterns of the n subscribers to a telephone system into the patterns in which the first person is not calling anyone else, and the patterns in which the first person is making a call. There are T(n − 1) connection patterns in which the first person is disconnected, explaining the first term of the recurrence. If the first person is connected to someone, there are n − 1 choices for that person, and T(n − 2) patterns of connection for the remaining n − 2 people, explaining the second term of the recurrence.
Summation formula and approximation
The telephone numbers may be expressed exactly as a
In each term of the first sum, gives the number of matched pairs, the
binomial coefficient counts the number of ways of choosing the elements to be matched, and the
In other words, the telephone numbers may be read off as the coefficients of the
Taylor series of exp(x2/2 + x), and the nth telephone number is the value at zero of the nth derivative of this function.
This function is closely related to the exponential generating function of the
Hermite polynomials, which are the
matching polynomials of the complete graphs.
The sum of absolute values of the coefficients of the nth (probabilist's) Hermite polynomial is the nth telephone number, and the telephone numbers can also be realized as certain special values of the Hermite polynomials:
For large values of n, the nth telephone number is divisible by a large
power of two, 2n/4 + O(1). More precisely, the
2-adic order (the number of factors of two in the
prime factorization) of T(4k) and of T(4k + 1) is k; for T(4k + 2) it is k + 1, and for T(4k + 3) it is k + 2.
For any prime number p, one can test whether there exists a telephone number divisible by p by computing the recurrence for the sequence of telephone numbers, modulo p, until either reaching zero or
detecting a cycle. The primes that divide at least one telephone number are
2, 5, 13, 19, 23, 29, 31, 43, 53, 59, ... (sequence A264737 in the
The odd primes in this sequence have been called inefficient. Each of them divides infinitely many telephone numbers.
abSolomon, A. I.; Blasiak, P.; Duchamp, G.; Horzela, A.; Penson, K.A. (2005), "Combinatorial physics, normal order and model Feynman graphs", in Gruber, Bruno J.; Marmo, Giuseppe; Yoshinaga, Naotaka (eds.), Symmetries in Science XI, Kluwer Academic Publishers, pp. 527–536,
^A direct bijection between involutions and tableaux, inspired by the recurrence relation for the telephone numbers, is given by Beissinger, Janet Simpson (1987), "Similar constructions for Young tableaux and involutions, and their application to shiftable tableaux", Discrete Mathematics, 67 (2): 149–163,