Schematic algebra - Introduction
| Moreau, Charles Paul Narcisse, 1837-1916 French soldier and mathematician. He dealt with combinatorial problems (chess tower) and games. He introduced a function for quantifying the number of necklace combinations. |
Combinatorics of necklaces
Most chords of a 12-tone system have 12 transpositions. But not all of them. Some have transpositions less (for example triad C5 + four transpositions, quad Cdim three transpositions ...) The resulting general combinatorial problem thus differs from other commonly studied cases and is usually included under the name "Combinatorics of Necklaces".
Charles Moreau (1872) was probably the first to calculate the number of necklace combinations. He used Mobius' function (function of counting necklaces, necklace polynomial) to express the count.
Later - in 1937, the Hungarian mathematician George Pólya published a general theory that allows to calculate various arrangements (symmetry, etc.). He so rediscovered the theory with which American mathematician John Howard Redfield had already come up about ten years earlier. The theory is known by the Polya's name and it uses the Euler's function to quantify the count.
However, structures can also be calculated more easily - without using Mobius or Euler's function. If we define the order of the system "k" as the number of elements (corals on the necklace ...), it is just necessary add to the combinatorics of the order "k" also all the subordinate orders "d", which divide the number "k". Such an addition seems to be natural and clear and shows the beauty of mathematical structures. From the number-formations (which we call G-systems) some known relations (Fermat's, Wilson's, Lerch's theorem, etc.) become obvious - at first glance (without long proofs).
At the same time, it is possible not only to count classes (species) but also to construct them - by an algorithm similar to the Eratostenes' sieve. Instead of multiples of prime numbers here came rotations (ie multiplication by the number "n" according to the specified modulus "r")..
Lyndon, Roger
| Lyndon, Roger Conant, 1917-1988, American mathematician. He dealt with logic, combinatorics, group theory and geometry. |
Lyndon's words
In G-systems theory, we use the term instance (any grouping) and the term classes (species, they includes all rotations). The representative of a given class is the g-number, that is the first (lowest) number of the class (in the order of generation). In mathematics, the term Lyndon's word was used for the g-number. This is defined as: "a non-empty string that is strictly smaller in lexicographic order than all its rotations" and thus corresponds practically exactly to what we hereinafter call the g-number.
Golomb, Solomon
| Golomb, Solomon Wolf, 1932-2016 American mathematician. He dealt with combinatorics, number theory and applied mathematics (coding, encryption, math games, etc.). |
Derivation of simple polynomials
The total numbers of self classes (ie not nested classes) correspond to the numbers of simple polynomials, as derived by Gauss…! How to derive simple polynomials from self classes (Lyndon's words) is not at first glance obvious (coefficients of polynomials do not resemble the relevant numbers of self classes ...)
But solutions already exist. Probably based on work S.W.Golomb (Irreducible polynomials, synchronization codes, primitive necklaces, and the cyclotomicalgebra, 1969).
Bshouty, Nader
| Bshouty, Nader H., Israeli mathematician. He works in the field of learning theory, Lyndon word enumeration algorithms,... He also studied theories of Galois extensions. |
Among newer publications, it is possible to find works of N.H.Bshouty (Enumerating all the Irreducible Polynomials over Finite Field, 2016).
Polynomials have to be computed by multiplying the expressions of the form (x-a) ...
Self classes
Numbers of types of musical structures with a given number of tones "L" for individual orders "k" determines the "triangle of self classes". It forms as if the core of Pascal's triangle. On its diagonals are sequences whose generating functions are described by H. Kociemba in connection with the Rubik's Cube. In these functions we can also observe a certain "nesting" (from lower orders) - as in whole G-systems.
One of the two main sequences on the "backbone" of the triangle is the Catalan sequence (The second sequence is probably a supplement to it ...!?).
The numbers on the individual lines show an interesting feature - a possibility resetting them. Just like Pascal's triangle can be zeroed by multiplication individual members in the rows by sequence {1, −1,1, −1, ...}, so it is possible zero triangle of self classes (own species) by sequence {1, −1,0,1, −1,0, ...} !?
So we can divide the numbers on the lines into three groups. Each member in a certain row is a member of some Kociemba's sequence. To enumerate these terms (of shifted diagonal sequences) in general for each of the three groups, does not seem to be completely trivial ...
But there is one more thing. When we divide the members in rows according to the numbering modulo 3 (as zeroing takes place using {1, −1,0}), we get the sums of A0, A1, A2, where we know that A0 = A1. But how much is A2 and how much does 3 * A0 = 3 * A1 differ from A1 + A2 + A3? The values A0 = A1 form the sequence P1: 0, 1, 1, 2, 3, 6, 10, 19, 33, 62, 112, 210, 387, 728, 1360, ... i.e. OEIS: A165920. Values A1 + A2 + A3 sequence P2: 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080 i.e. OEIS: A001037. The non-zero differences 3 * P1 - P2 again form the sequence 0, 1, 1, 2, 3, 6, 10, 19, 33, 62, ... i.e. OEIS: A165920! In other words - the sequence P2 is three times P1 reduced by a sequence whose every third member is an element P1 (and other members are zero).
Follow - up considerations
Just as types of musical structures are constructed, it is also possible to construct types in other number systems, such as those studied by Mersenne, Fermat, Kummer, or Wieferich. Here it only depends on the module "r", according to which the number system (R-system) is built. In general, these systems have already been studied by Gauss - who has shown how such systems are nested and how their numbers of species can be accurately calculated.