A284039 Wiener index of the n-permutation star graph.

0, 1, 27, 744, 26520, 1239840, 74662560, 5663831040, 530098007040, 60105991680000, 8127440487936000, 1292894601191424000, 239129895342514176000, 50899158690744139776000, 12356174324714508288000000, 3393918280427832764006400000, 1047355019625604129593753600000

Offset: 1

Author

Eric W. Weisstein, Sep 13 2017

Keywords

• nonn

Comments

The permutation star graph of order n is a vertex transitive graph with n! vertices and degree n-1. The graph can be constructed as the Cayley graph of the permutations of 1..n with the n-1 generators (1 2), (1 3)..(1 n) where (1 k) is the transposition of 1 and k. The number of nodes at distance k from a specified node is given by A007799(n,k). - Andrew Howroyd, Sep 17 2017

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Permutation Star Graph
• Eric Weisstein's World of Mathematics, Wiener Index

Crossrefs

Cf. A007799, A192837.

Programs

Mathematica

a[n_, 0] = 1;
a[n_, 1] = n - 1;
a[n_, 2] = (n - 1) (n - 2);
a[n_, k_ /; k >= 2] := a[n, k] = (n - 1) a[n - 1, k - 1] + Sum[j a[j, k - 3], {j, 1, n - 2}];
Table[n!/2 Sum[k a[n, k], {k, Floor[3 (n - 1)/2]}], {n, 10}]

Formula

a(n) = n!/2 * Sum_{k=1..floor(3*(n-1)/2)} k*A007799(n, k). - Andrew Howroyd, Sep 17 2017

Extensions

Terms a(8) and beyond from Andrew Howroyd, Sep 17 2017