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Also simple graphs containing a crossing pair of edges, where two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.

Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.

This is the third member, q=3, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1)), A029635 (q=2) (but with T(0,0)=2, not 1).

This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column no. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x) = Sum_{m=0..n} T(n,m)*x^m is G(z,x) = g(z)/(1-x*z*f(z)). Here: g(x) = (3-2*x)/(1-x), f(x) = 1/(1-x), hence G(z,x) = (3-2*z)/(1-(1+x)*z).

The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} T(n-1-k,k) = A000285(n-2), n>=2, with n=1 value 3. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Central terms: T(2*n,n) = A028329(n) = A100320(n) for n > 0, A028329 are the central terms of triangle A028326. - Reinhard Zumkeller, Apr 08 2012

Let P be Pascal's triangle, A007318 and R the Riordan array, A097805. Then Pascal triangle (1,q) = ((q-1) * R) + P. Example: Pascal triangle (1,3) = (2 * R) + P. - Gary W. Adamson, Sep 12 2015

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.

Capturing is a weaker condition than nesting. A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. For example, {{1,3,5},{2,4}} is capturing but not nesting, so is counted under a(5).

Principal diagonal: A002418

Antidiagonal sums: A005585

row 1, (1,3,6,...)**(1,2,3,...): A000332

row 2, (1,3,6,...)**(2,3,4,...): A005582

row 3, (1,3,6,...)**(3,4,5,...): A095661

row 4, (1,3,6,...)**(4,5,6,...): A095667

For a guide to related arrays, see A213500.

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 set partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y, and capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

The set of maximal elements of an ideal is an antichain; conversely, the down-set of a nonempty antichain is an ideal. The down-set of the top element of the n-dimensional Boolean lattice contains all 2^n elements of the lattice, and thus is not a proper ideal.

Empirically, the rows are unimodal.

By the Markowsky paper, T(n, k) = T(n, 2^n - k).

Also, T(n,k) is the number of n-dimensional Ferrers diagrams with k nodes (i.e., (n-1)-dimensional partitions) that fit into an n-dimensional hypercube of side 2 (i.e., a Boolean or binary hupercube). T(n, k) = T(n, 2^n - k) follows from the map that takes a Ferrers diagram to its complement in the box. - Suresh Govindarajan, Apr 10 2016

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.

The Kneser graph K(n,2) is the graph whose vertices represent the 2-subsets of {1,2,...,n} and two vertices are connected if and only if they correspond to disjoint subsets.

K(n,2) is disconnected for n<=4.

K(5,2) is the Petersen graph.

The Kneser graph K(n,2) is distance-regular with intersection array [(n-2)*(n-3)/2, 2*(n-4); 1, (n-3)*(n-4)/2].