A296190 Numerators of Harary index for the n-permutation star graph.

0, 1, 10, 123, 2202, 59040, 2287680, 121394000, 92649740400, 105538103163360, 1034297134668000, 134399089883282400, 27076064087538702720, 5451799851068349018240, 19300076847195336557164800, 4599598343095846092562560000, 1682634821690958905899793664000

Offset: 1

Author

Eric W. Weisstein, Dec 07 2017

Keywords

• nonn
• frac

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, Dec 09 2017

Links

• Eric Weisstein's World of Mathematics, Harary Index
• Eric Weisstein's World of Mathematics, Permutation Star Graph

Crossrefs

Cf. A296057 (denominators), A007799, A284039.

Programs

Mathematica

A007799[n_, i_] := Sum[Binomial[n - 1, k] Binomial[n - 1 - k, t] StirlingS1[k + 1, i - k + 1 - 2 t] (-1)^(i + 2 - t), {k, 0, Min[n - 1, i + 1]}, {t, Max[0, Ceiling[(i - 2 k)/2]], Min[n - 1 - k, Floor[(i + 1 - k)/2]]}];
Table[n! Sum[A007799[n, k]/k, {k, Floor[3 (n - 1)/2]}]/2, {n, 20}] // Numerator (* Eric W. Weisstein, Dec 09 2017 *)

Formula

a(n)/A296057(n) = (n!/2) * Sum_{k=1..floor(3*(n-1)/2)} A007799(n, k)/k. - Andrew Howroyd, Dec 09 2017

Extensions

a(9)-a(17) from Andrew Howroyd, Dec 09 2017