Some mathematical problems I can not solve
                                                   by V. Mazorchuk

1. (authors: Oleksandr Ganyushkin and  Volodymyr Mazorchuk) Let n be a positive integer and  N={1,2,..,n}.
Denote by a_n  the number of vectors (A_0,A_1,...,A_n) such that A_i is a subset of N and |A_i|=i  for all
i=1,2,...,n; and  such that the collection {A_0,A_1,...,A_n}  is  closed with respect to intersection (i.e. for each
i and j the intersection of A_i and A_j has the form A_s for some s).  Question: find a closed  formula for a_n.

Origin of the problem: a_n is the number of cross-sections of Green's relation D, which consist of idempotents,
                                    in the full symmetric inverse semigroup IS_n.

Reference: O. Ganyushkin, V.Mazorchuk, L- and R- cross-sections of IS_n, Communications in Algebra,
Vol. 31, No. 9, pp. 4507-4523, 2003. (see Problem 2, page 4508)

2. (authors: Ganna  Kudryavtseva and Volodymyr Mazorchuk) Let n be a positive integer. Consider the
alphabet {1,2,..,n} with the natural order on letters. Denote by b_n the number of words in this alphabet which
satisfy the following condition: Between any two occurrences of the same letter there should be an occurence
of a  smaller letter and an occurence of a bigger latter (for example, the word 2132 satisfies this condition
since we have 1<2 and 3>2 occuring between the two 2's, whereas the word 2432 does not satisfy  this
condition as there is no letter smaller than 2 occurring between the two 2's). It is very easy to show  that b_n
is finite. Question: find a closed  formula for b_n.

Calculations using Maple give the following table:

n        1    2     3       4        5           6
b_n    2    5   18   115  1710   83973


Origin of the problem:  b_n is the cardinality of Kiselman's semigroup K_n.

Reference:  G.Kudryavtseva, V.Mazorchuk, On Kiselman's semigroup, math.GR/0511374. (see Theorem 6)

3. (author: Eva Kopecka) Does there exist a finite number k such that the following condition is satisfied: for any
small  e>0 there exist vectors v_1,..., v_k in the infinite-dimensional separable Hilber space l_2 and a piecewise
linear path from the point A=(1,0,0,0,...) to the point B=(0,1-e,0,0,..) (both points are  in l_2) such that each linear
segment of the path is parallel to one of the vectors v_1,..., v_k, and that going along the path from A to B the
distance to the origin monotonically decreases?

4. (author: Vasyl Ostrovskyi) Does there exist a finite number k such that for any small e>0 there exists a
collection of k straight lines l_1,...,l_k in R^2 such that there is a path starting at the point (0,0) and terminating
at the point (1,0) such that any segment of the path is obtained as the orthogonal projection of the endpoint
of the previous segment onto one of our lines and such that the sum of the squares of the lengths of all
these segments is smaller than e?

Any hint for a solution (or simply a solution) to any of the above problems would be greatly appreciated.