cp's OEIS Frontend

Equivalently, number of connected unicyclic (i.e., containing one cycle) graphs on n labeled nodes. - Vladeta Jovovic, Oct 26 2004

a(n) is the number of trees on vertex set [n] = {1,2,...,n} rooted at 1 with one marked inversion (an inversion is a pair (i,j) with i > j and j a descendant of i in the tree). Here is a bijection from the title graphs (on [n]) to these marked trees. A title graph has exactly one cycle. There is a unique path from vertex 1 to this cycle, first meeting it at k, say (k may equal 1). Let i and j be the two neighbors of k in the cycle, with i the larger of the two. Delete the edge k<->j thereby forming a tree (in which j is a descendant of i) and take (i,j) as the marked inversion. To reverse this map, create a cycle by joining the smaller element of the marked inversion to the parent of the larger element. a(n) = binomial(n-1,2)*A129137(n). This is because, on the above marked trees, the marked inversion is uniformly distributed over 2-element subsets of {2,3,...,n} and so a(n)/binomial(n-1,2) is the number of trees on [n] (rooted at 1) for which (3,2) is an inversion. - David Callan, Mar 30 2007

a(n) is the number of simple labeled graphs with n nodes and n edges. - Geoffrey Critzer, Nov 02 2014

These graphs are not necessarily covering, but the covering case is A367863, unlabeled A006649, and the unlabeled version is A001434. - Gus Wiseman, Dec 22 2023

The majority of those graphs of order 4 are trees since we have 16 trees and only 9 unicycles. See example.

Also connected graphs covering n vertices with at most n edges. The unlabeled version is A005703. - Gus Wiseman, Feb 19 2024

Unlike the connected case (A057500), these graphs may have more than one cycle; for example, the graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}} has multiple cycles.

Number of n x n binary matrices without zero rows and with distinct rows up to permutation of rows, cf. A014070.

Row 0 of square array A136555.

From Gus Wiseman, Dec 19 2023: (Start)

Also the number of n-element sets of nonempty subsets of {1..n}, or set-systems with n vertices and n edges (not necessarily covering). The covering case is A054780. For example, the a(3) = 35 set-systems are:

{1}{2}{3} {1}{2}{12} {1}{2}{123} {1}{12}{123} {12}{13}{123}

{1}{2}{13} {1}{3}{123} {1}{13}{123} {12}{23}{123}

{1}{2}{23} {1}{12}{13} {1}{23}{123} {13}{23}{123}

{1}{3}{12} {1}{12}{23} {2}{12}{123}

{1}{3}{13} {1}{13}{23} {2}{13}{123}

{1}{3}{23} {2}{3}{123} {2}{23}{123}

{2}{3}{12} {2}{12}{13} {3}{12}{123}

{2}{3}{13} {2}{12}{23} {3}{13}{123}

{2}{3}{23} {2}{13}{23} {3}{23}{123}

{3}{12}{13} {12}{13}{23}

{3}{12}{23}

{3}{13}{23}

Of these, only {{1},{2},{1,2}}, {{1},{3},{1,3}}, and {{2},{3},{2,3}} do not cover the vertex set.

(End)

Number of labeled loop-graphs with n vertices and at most n edges. - Gus Wiseman, Feb 14 2024

Row-sums of left portion of A369199.

Also, number of n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.