cp's OEIS Frontend

These are digraphs with the property that, if the edges are listed in lexicographic order, the sequence of targets is weakly increasing. For example, the digraph with lexicographically ordered edge set {(1,2),(2,1),(3,1),(3,2)} is nesting because the targets are (2,1,1,2), a sequence that is not weakly increasing.

Also the number of non-semicrossing digraphs with vertices {1..n}, where two edges (a,b), (c,d) are semicrossing if a < c and b < d or a > c and b > d. For example, the a(2) = 4 non-semicrossing digraph edge-sets are:

{}

{11}

{12}

{21}

{22}

{11,12}

{11,21}

{12,21}

{12,22}

{21,22}

{11,12,21}

{12,21,22}

Apparently a duplicate of A152254. - R. J. Mathar, Jul 12 2019

If an integer N is squarefree and has n+2 distinct prime factors then a(n) is the number of product signs needed to write the factorizations of N, so a(n)=A076277(N). - Floor van Lamoen, Oct 17 2002

Convolved with powers of 2 = A058681: (1, 7, 36, 171, 813, ...). Cf. triangle A180338. - Gary W. Adamson, Aug 28 2010

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.

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).

From Petros Hadjicostas, Oct 10 2019: (Start)

Mallows (1979) comments that I. J. Good did not consider all kinds of independence between n random variables in deriving his formula for the e.g.f. of a(n). Unfortunately, the paper by Good (1975), where the proof of the e.g.f. was published, is not easily accessible.

We give a sketch of a proof of the formula below (using only the ideas of independence that I. J. Good considered). Given r.v.'s X_1, X_2, ..., X_n, we choose s of them on which to condition (where s = 0, 1, 2, ..., n-2). By "condition", we mean that we condition on every possible s-tuple of values of those chosen variables. This can be done in C(n, s) ways.

Note that we can only condition on up to n-2 variables, because we need at least two variables to define any kind of independence: conditional (s >= 1) or unconditional (s = 0). Thus, 2 <= n-s <= n.

From the remaining n-s variables, we choose t of them (where 2 <= t <= n-s) on which we will define (or test) independence. [According to Mallows (1979), this is the only kind of independence Good (1975) considers.] There are C(n-s, t) ways to choose the t variables on which to define (or test) independence.

Now, there are Bell(t) - 1 = A058692(t) ways to partition the set of t chosen variables into one or more subsets, say {S_1, ..., S_r} (the order of the subsets is not important). Here |S_i| >= 1 and (S_1 union S_2 union ... union S_r) equals the t chosen variables. Thus, there are Bell(t) - 1 ways to factor the joint p.d.f. of the chosen t variables into the product of the joint marginal p.d.f.'s of the variables in S_i (i = 1, ..., r). [By "joint marginal p.d.f." for group S_i we mean the joint p.d.f of all the variables in group S_i.] Each such factorization defines a different kind of independence of the chosen variables (conditional on the original chosen s variables).

Thus, in the sense of Good (1975, 1979), there are a(n) = Sum_{s = 0..n-2} Sum_{t = 2..n-s} C(n, s) * C(n-s, t) * (Bell(t) - 1) kinds of independence among the variables.

Letting k = n-s, we get a(n) = Sum_{k = 2..n} Sum_{t = 2..k} C(n, n-k) * C(k, t) * (Bell(t) - 1) = Sum_{k = 2..n} Sum_{t = 2..k} C(n, k) * C(k, t) * (Bell(t) - 1).

The second formula for a(n) follows from the fact that Sum_{m = 2..k} binomial(k,m) * (Bell(m) - 1) = A058681(k) = Bell(k+1) - 2^k.

Counting all kinds of independence, as suggested by Mallows (1979), seems to be a very difficult task. For example, for n = 3, he finds a(3) = 17 kinds of independence rather than 10.

(End)

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

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

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.

A006905(n) = Sum_{k=0..n} A001035(k) * T(n, k). - Michael Somos, Jul 18 2021

T(n, k) is the number of idempotent relations R on [n] containing exactly k strongly connected components such that the following conditional statement holds for all x, y in [n]: If x, y are in distinct strongly connected components of R then (x, y) is not in R. - Geoffrey Critzer, Jan 10 2024

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

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