Number of permutations of [n] with 2 sequences.
Number of 2 X n binary matrices that avoid simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in same relative order as those in the triple (x,y,z). -
Sergey Kitaev, Nov 11 2004
The number of edges in the dual Edwards-Venn diagram graph with n-1 digits when n>2.
a(n) (n>=6) is the number of vertices in the molecular graph NS2[n-5], defined pictorially in the Ashrafi et al. reference (Fig. 2, where NS2[2] is shown). -
Emeric Deutsch, May 16 2018
With regard to the comment above about a(n) being the "number of permutations of [n] with 2 sequences", we refer to Ex. 13 (pp. 260-261) of Comtet (1974), who uses the definition of a "séquence" given by André in several of his papers in the 19th century.
In the terminology of array
A059427, these so-called "séquences" in permutations (defined by Comtet and André) are called "alternating runs" (or just "runs"). We discuss these so-called "séquences" below.
If b = (b_1, b_2, ..., b_n) is a permutation of [n], written in one-line notation (not in cycle notation), a "séquence" in a permutation of length l >= 2 (according to Comtet) is a maximal interval of integers {i, i+1, ..., i+l-1} for some i (where 1 <= i <= n-l+1) such that b_i < b_{i+1} < ... < b_{i+l-1} or b_i > b_{i+1} > ... > b_{i+l-1}. (The word "maximal" means that, in the first case, we have b_{i-1} > b_i and b_{i+l} < b_{i+l-1}, while in the second case, we have b_{i-1} < b_i and b_{i+l} > b_{i+l-1} provided b_{i-1} and b_{i+l} can be defined.)
When defining a "séquence", André (1884) actually refers to the list of terms (b_i, b_{i+1}, ..., b_{i+l-1}) rather than the corresponding index set {i, i+1, ..., i+l-1} (which is essentially the same thing).
For more details about these so-called "séquences" (or "alternate runs"), see the comments and examples for sequence
A000708.
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For n >= 1, a(n+2) is the number of shortest paths from (0,0) of a square grid to {(x,y): |x|+|y| = n} along the grid line. -
Jianing Song, Aug 23 2021
