1. Introduction
Discovery of n-cube combinatoric properties remains a relevant topic, which extends the connections of mathematical fields [1] -[4] . The bijective mappings play an important role in enumerative combinatorics as broad alighted in classical works of G.-C. Rota and R. P. Stanley [5] [6] . Bijective form for some constructive world [7] could be considered not only as suitable for enumerative problems, but also with point of view of effective computing synthesis (algorithms and operations with the potential large parallelism) in such frame. Such approach is considered in the article on the base of constructions (computing) for k-paths as complexes of k-faces in n-cube.
2. Shortly on Cubants
One of bijections for k-faces of n-cube was proposed in [8] . Let be
—reper in
, alphabet 
and the set 
of all n-digits words. So some word of the set is
. Each k-face can be represent as Cartesian product 
of unit-segments 
for 
and translation 
along the rest basis such, that
. So the bijection for k-face 
can be written as next:
where 
for 
and 
for
. 
for translation along 
and
, when translation is out. Such representation allows to store traditional coding for n-cube vertex coding (vertice is 0-face). Let character 
be supplement and then
. Character-oriented operation multiplication (intersection) is determined on 
(all quaternary n-digital words) with next rules:
Really it’s intersection of sets: “0, 1”—endpoints of unit-segment and “2” corresponds full unit-segment. For short all words from 
are titled as cubants. So we can say the set of cubants forms monoid with unit-cubant 
(n-face in n-cube, i.e. itself n-cube).
The character-oriented operation of addition for cubants is prescribed by next rules:
Result of the operation is cubant for convex hull face and therefore one can write:
.
Short-list of operations on cubants is outlined below:
1)
—counting of character 
in cubant
. Result is from
.
2)
—exchanging of all “0” to “1” and all “1” to “0” in cubant
. Result 
is cubant for antipodal (a.p.) face.
3)
—operation multiplication. Result is cubant 
for common face, if
. In case 
it’s
—length of shortest path along edges between faces with cubants 
and
, in accordance with (1).
4)
. Result is cubant 
in accordance with (2).
5)
. Exchanging letters “2” on “0” in such 
of
, for which
, and “2” on “1”, for which
. Result 
has got properties 
and
.
6)
.
Calculation of Hausdorff-Hamming (HH) distance for faces with cubants
,
[9] .
7)
—boundary for face with cubant
. Result is a set of cubants corresponding the all hyperfaces.
Algorithm of HH-distance calculation was proposed in [9] and all k-faces of n-cube form finite metric HH-space. Simplicity of the algorithm gives foundation to add it to operations for cubants. By the way the same algorithm realized calculation of Gromov-Hausdorff (GH) distance between cubes (as finite metric spaces) of different dimensions.
3. Matrix Representation of k-Path
Below we consider complexes of k-faces (here k-dimension of face in contrast to [4] , where k-length along edges as shortest paths between vertex). Now we will give definition of k-path between two of antipodal (a.p.) vertices in terminology of cubants. No limits of common we can consider cubants 
and 
for a.p. vertices. Then the set of cubants
, 
is bijectivial form of shortest k-path such, that next conditions are satisfied for
:
We represent set of such cubants in more visible form of 
matrix
:
It’s easy to check next matrix corresponds to k-path for available 
and
:
The columns of the matrix are denoted by
. Then available permutation of columns stores satisfying of conditions (3) and
. All such matrices (under permutations from symmetric group
) represent the isomorphic k-paths with partition
:
. For case (4):
Evidently the matrices with different partitions correspond non-isomorphic k-paths. Therefore we can define the such partitions as numerical invariants, which allow one to distinguish among non-isomorphic k-paths, i.e. to classify k-paths. Now we must remark a specific property of
. Here the columns are written as horizontal rows. So each 
can have view only of four types:
Roughly speaking the sequence of the same characters in 
denies “gaps”, since otherwise the condition of 
is not satisfied.
The specific property leads to situation, when some partitions are not represented in frame of T. For example the number of non-isomorphic k-paths classes 
for
, 
is equal 4, though
,
:
, 
, 
, 
, 
At that time
,
:
, 
, 
, 
, 
, 
So
.
Now we consider common form of 
matrix 
of special type (conditions (3) are satisfied):
Number of vertical columns with “2” (VC) can lay in interval from 0 to 
and each of them corresponds to “2-stairs” (SC) from 
to 1, for
. The case 
must be analyze separately. So 
for 
and common bounds of classes number are next:
Now about case
. Set SC columns includes such, which have 
character 2, i.e. coincide with one of VC-column. The number of such SC columns is equal to
. Therefore:
One can combine (6) and (7) in single result:
One can give title the staircase for 
of type (5).
We considered above k-paths for antipodal (a.p.) vertices 
and
. Now let available two vertices in n-cube are given and hamming distance between them is equal to
. Then computing of matrix 
for k-path is reduced to a.p. case. Therefore we delete in pares the same 
digits. So the rest 
digits correspond a.p. vertices in face-convex hull for these cut vertices. Our previous techniques may be successfully here with addition of deleted 
digits in columns of
. Shortly speaking the sequence of steps looks like this: extraction of a.p. part in given vertices (deleting of differing in pairs digits)
the choice of matrix 
of type (5)
inserting of columns 
with deleted digits in
.
More general problem is to construct of k-path, when two a.p. vertices
, 
and k-face 
are given
. Without loss of generality let left digit of 
is “2”. So the first row of matrix 
is
. Algorithm consists of sequential generations
, which follows 
in matrix
. For case 
and 
we assign 
and shift character 2 in nearest digit
, for which
. In common case if such digit in 
is absent, the procedure is completed. Let here 
then we assign for digits 
character “2” and the same characters from 
for
.
In common case for
,
we produced in analogous fashion, beginning with duplicating in 
the same characters of 
before first meeting “2”. Then we assign “1” for next digit of 
and further digits are determined in accordance with rules for
. One can represent the digit-wise rules as next scheme:
One can give title of the procedure as pressing characters “2” with single inversion 0 - 1.
Examples of 2-paths in 6-cube is represented step by step below (Figure 1).
Figure 1. 2-paths (T1, T2) are drawn onto flat projection of 2-skeletone of 6-cube.
HH-distance may be taken in account constructing some k-paths (operation 6)). So 
table 
for 
and 
(2-paths in 6-cube) is following:
, 
, 
It follows: 
(Figure 1).
To remark although our exposition is short, the most of operations for cubants are realized digitwise, i.e. in parallel. It’s clearly visible, if we’ll use for computer the bitwise mapping
, 
, 
,
.
4. Conclusions
In conclusion, we give the main statement of the article.
Minimal number s of k-faces in k-path between a.p. vertices in n-cube is equal to
. The bounds for number of non-isomorphic k-path classes are
, where 
are partitions integer 
in 
parts with constraint 
for maximal part. Lower bound 
is always realized by staircase matrix.