On Classification of k-Dimension Paths in n-Cube

The shortest k-dimension paths (k-paths) between vertices of n-cube are considered on the basis a bijective mapping of k-faces into words over a finite alphabet. The presentation of such paths is proposed as ( ) 1 n k n − + × matrix of characters from the same alphabet. A classification of the paths is founded on numerical invariant as special partition. The partition consists of n parts, which correspond to columns of the matrix.


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.

Shortly on Cubants
One of bijections for k-faces of n-cube was proposed in [8].Let be { } .Character-oriented operation multiplication (intersection) is determined on * n A ′ (all quaternary n-digital words) with next rules: 0 0 0, 0 1 1 0 , 0 2 2 0 0, Really it's intersection of sets: "0, 1"-endpoints of unit-segment and "2" corresponds full unit-segment.For short all words from * n A ′ are titled as cubants.So we can say the set of cubants forms monoid with unit-cubant 22 2  (n-face in n-cube, i.e. itself n-cube).

7) D
∂ -boundary for face with cubant D .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.

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 00...0 and 11...1 for a.p. vertices.Then the set of cubants { }  ( )( ) We represent set of such cubants in more visible form of n s × matrix T : The columns of the matrix are denoted by , 1, , ( ) .
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 among non-isomorphic k-paths, i.e. to classify k-paths.Now we must remark a specific property of 202000 201200 221100 121120 121112 Now we consider common form of n s × matrix T of special type (conditions (3) are satisfied): One can combine ( 6) and ( 7) in single result: One can give title the staircase for T of type (5).
We considered above k-paths for antipodal (a.p.) vertices 00 0  and 11 1  .Now let available two vertices in n-cube are given and hamming distance between them is equal to ( ) 0 r r n < < .Then computing of matrix T for k-path is reduced to a.p. case.Therefore we delete in pares the same n r − digits.So the rest r digits correspond a.p. vertices in face-convex hull for these cut vertices.Our previous techniques may be successfully here with addition of deleted n r − digits in columns of j D * .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 T of type (5) → inserting of columns j D * with deleted digits in T .More general problem is to construct of k-path, when two a.p. vertices 1 00 0 v =  , 2 11 1 v =  and k-face ( ) D are given ( ) 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).HH-distance may be taken in account constructing some k-paths (operation 6)).So

Conclusions
In all n-digits words.So some word of the set is1 , , n D d d = .Each k-face can be represent as Cartesian product ( ) ∏ of unit-segments ( ) 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

) 1 2 D D × -operation multiplication. Result is cubant 3 D
for common face, if ( ) length of shortest path along edges between faces with cubants 1 s easy to check next matrix corresponds to k-path for available n and k : available permutation of columns stores sa- tisfying of conditions (3) and ( ) matrices (under permutations from symmetric group n S )represent the isomorphic k-paths with partition λ :

jD
*  .Here the columns are written as horizontal rows.So each j D * can have view only of four typesRoughly speaking the sequence of the same characters in j D * denies "gaps", since otherwise the condition of min s 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
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 00 ∅ → , 0 01 → , 1 10 → , 2 11 → .
loss of generality let left digit of 1 D is "2".So the first row of matrix T is 1 D .Algorithm consists of sequential generations 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 1 n k − + .The bounds for number of non-isomorphic k-path classes are for maximal part.Lower bound k is always realized by staircase matrix.