^{1}

^{*}

^{1}

^{*}

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 (*n* － *k* + 1)×*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.

Discovery of n-cube combinatoric properties remains a relevant topic, which extends the connections of mathematical fields [

One of bijections for k-faces of n-cube was proposed in [

where

Really it’s intersection of sets: “0, 1”—endpoints of unit-segment and “2” corresponds full unit-segment. For short all words from

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)

2)

3)_{ }in accordance with (1).

4)

5)

6)

Calculation of Hausdorff-Hamming (HH) distance for faces with cubants

7)

Algorithm of HH-distance calculation was proposed in [

Below we consider complexes of k-faces (here k-dimension of face in contrast to [

We represent set of such cubants in more visible form of

It’s easy to check next matrix corresponds to k-path for available

The columns of the matrix are denoted by

Roughly speaking the sequence of the same characters in

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

At that time

So

Now we consider common form of

Number of vertical columns with “2” (VC) can lay in interval from 0 to

Now about case

One can combine (6) and (7) in single result:

One can give title the staircase for

We considered above k-paths for antipodal (a.p.) vertices

More general problem is to construct of k-path, when two a.p. vertices

In common case for

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 (

HH-distance may be taken in account constructing some k-paths (operation 6)). So

It follows:

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

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