cp's OEIS Frontend

Perturbation expansion in quantum field theory: spinor case in 4 spacetime dimensions.

a(n)*2^(-n) is the coefficient of the x^(2*n-1) term in the series reversal of the asymptotic expansion of 2*DawsonF(x) = sqrt(Pi)*exp(-x^2)*erfi(x) for x -> inf. - Vladimir Reshetnikov, Apr 23 2016

The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

A set partition is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y. Then a(n) is the number of topologically connected 2-uniform set partitions of {1...2n}. See my links for examples. - Gus Wiseman, Feb 23 2019

From Julien Courtiel, Oct 09 2024: (Start)

a(n) is the number of rooted bridgeless combinatorial maps with n edges (genus is not fixed). A map is bridgeless if it has no edge whose removal disconnects the graph. For example, for n=2, there are 4 bridgeless maps with 2 edges: 2 planar maps with 1 vertex (either two consecutive loops, or two nested loops), 1 toric map with 1 vertex, and 1 planar map with 2 vertices connected by a double edge.

Also, a(n) is the number of trees with n edges equipped with a binary tubing. A tube is a connected subgraph. A binary tubing of a tree is a nested set collection S of tubes such that 1. S contains the tube of all vertices 2. Every tube of S is either reduced to one vertex, or it can be can partitioned by 2 tubes of S.

(End)

Related to Schröder's second problem.

A001006 gives number of ways of drawing any number of nonintersecting chords between n points on a circle, while this sequence gives number of ways of drawing noncrossing chords between n points on a circle. The difference is that nonintersection chords have no point in common, while noncrossing chords may share an endpoint. - David W. Wilson, Jan 30 2003

For n>0, a(n) = number of lattice paths from (0,0) to (n-1,n-1) that consist of steps (i,j), i,j nonnegative integers not both 0 and that stay strictly below the line y=x except at their endpoints. For example, a(3)=8 counts the paths with following step sequences: {(2, 2)}, {(2, 1), (0, 1)}, {(2, 0), (0, 2)}, {(2, 0), (0, 1), (0, 1)}, {(1, 0), (1, 2)}, {(1, 0), (1, 1), (0, 1)}, {(1, 0), (1, 0), (0, 2)}, {(1, 0), (1, 0), (0, 1), (0, 1)}. If the word "strictly" is replaced by "weakly", the counting sequence becomes A059435. - David Callan, Jun 07 2006

The nodes on the circle are distinguished by their positions but are otherwise unlabeled. - Lee A. Newberg, Aug 09 2011

From Gus Wiseman, Jun 22 2019: (Start)

Conjecture: Also the number of simple graphs with vertices {1..n} not containing any pair of nesting edges. Two edges {a,b}, {c,d} where a < b and c < d are nesting if a < c and b > d or a > c and b < d. For example, the a(0) = 1 through a(3) = 8 non-nesting edge-sets are:

{} {} {} {}

{12} {12}

{13}

{23}

{12,13}

{12,23}

{13,23}

{12,13,23}

Cf. A001519, A117662, A326244, A326256, A326257, A326279.

(End)

A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y. - Gus Wiseman, Feb 19 2019

Two edges {x,y}, {z,t} are crossing if either x < z < y < t or z < x < t < y. If the vertices are arranged in a circle, this is equivalent to crossing of chords.

Covering means there are no isolated vertices.

A set partition is crossing if it contains a pair of blocks of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

The topologically connected components of a set partition correspond to the blocks of its minimal non-crossing coarsening.

Sequence terms are given by [x^n] ( (1 + x)^(k+2)/(1 - x)^k )^n for k = 1. See the crossreferences for related sequences obtained from other values of k. - Peter Bala, Sep 29 2015

Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for n >= 0. This is the case a = 1, b = 0. - Peter Bala, Aug 28 2016

a(n) is the number of graphs on vertices 1,...,n such that, when these vertices are arranged counterclockwise around a circle and edges are drawn as straight line segments, the resulting diagram is connected. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

In this interpretation, both intersecting (set theoretically) and crossing (topologically) edges are considered connected. - Gus Wiseman, Feb 23 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

A multiset is normal if its union is an initial interval of positive integers.

A multiset partition is crossing if it has a 2-element submultiset of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.

Covering means there are no isolated vertices.