cp's OEIS Frontend

a(n) is the number of Dyck paths of semilength n with no UUDD. See A025242 for a bijection between paths avoiding DDUU versus UUDD.

Also number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k>=1. - Alois P. Heinz, Oct 07 2015

a(n) is the number of bargraphs of semiperimeter n (n>=2). Example: a(4) = 5; the 5 bargraphs correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3]. - Emeric Deutsch, May 20 2016 [a(n) are the row sums of A271942 for n >= 2. Peter Luschny, Oct 18 2020]

a(n) is the number of skew Motzkin paths of length n. A skew Motzkin path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down), F=(1,0) (flat) and A=(-1,1) (anti-down) so that down and anti-down steps do not overlap. - Sergey Kirgizov, Oct 03 2018

From Gus Wiseman, Jul 04 2019: (Start)

Conjecture: Also the number of maximal simple graphs with vertices {1..n} and no weakly nesting edges. Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d. For example, the a(1) = 1 through a(5) = 13 edge-sets are:

{} {12} {13} {14} {15}

{12,23} {12,24} {12,25}

{13,24} {13,25}

{13,34} {14,25}

{12,23,34} {14,35}

{14,45}

{12,23,35}

{12,24,35}

{12,24,45}

{13,24,35}

{13,24,45}

{13,34,45}

{12,23,34,45}

Cf. A006125, A054726, A117662, A326244, A326257, A326289, A326293, A326329, A326337, A326338, A326340.

(End)

a(n) is the number of Dyck n-paths in which no nonterminal descent has the same length as the preceding ascent. Example: a(3) = 2 counts UUDUDD and UUUDDD where the latter path qualifies because DDD is the terminal descent. - David Callan, Dec 14 2021

Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d. A graph has its weakly nesting edges connected if the graph whose vertices are the edges and whose edges are weakly nesting pairs of edges is connected.

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.

Appears to be essentially the same as A003946.

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.

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.

Covering means there are no isolated vertices. 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.

Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d.

Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d.