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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 nesting if a < c < d < b or c < a < b < d. A graph has its nesting edges connected if the graph whose vertices are the edges and whose edges are nesting pairs of edges is connected.

Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d. A graph has its nesting edges connected if the graph whose vertices are the edges and whose edges are nesting pairs of edges is connected.

Two blocks are nesting if they are of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. A set partition has its nesting blocks connected if the graph whose vertices are the blocks and whose edges are nesting pairs of blocks is connected.

Two blocks are capturing if they are of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. A set partition has its capturing blocks connected if the graph whose vertices are the blocks and whose edges are capturing pairs of blocks is connected.

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.