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[[File:Gal.2.13.1.png|180px|right|link=http://www.teherba.org/OEIS-mat/coors/Test.svg]]
==Generating functions for coordination sequences of uniform tilings==
==Generating functions for coordination sequences of uniform tilings==
For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms.
For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a '''[http://www.teherba.org/OEIS-mat/coors/evalgf.txt tab-separated file]''' with the result.


For example:
As an example we examine the two lines (coseq.s) for '''Gal.2.13''' in this file:
  [https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]
  [https://oeis.org/A250120/a250120.html#2.13 2-Uniform Tiling 13 of 20] Standard notation: [3^6; 3^2.4.12]
  [https://oeis.org/A301692 A301692] Gal.2.13.1
  [https://oeis.org/A301692 A301692] Gal.2.13.1
O.g.f.: (-x^13-2*x^12+3*x^10+5*x^9+8*x^8+11*x^7+13*x^6+14*x^5+12*x^4+9*x^3+7*x^2+4*x+1) / (x^10-2*x^5+1)
  O.g.f.: (-x^13-2*x^12+3*x^10+5*x^9+8*x^8+11*x^7+13*x^6+14*x^5+12*x^4+9*x^3+7*x^2+4*x+1) / (x^10-2*x^5+1)
Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1  a14>s11
  Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1  a14>s11
Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2
  Denom. factorized: (x^5-1)^2
  [https://oeis.org/A301690 A301690] Gal.2.13.2  
  [https://oeis.org/A301690 A301690] Gal.2.13.2  
O.g.f.: (x^10+6*x^9+6*x^8+9*x^7+12*x^6+16*x^5+12*x^4+9*x^3+6*x^2+6*x+1) / (x^10-2*x^5+1)
  O.g.f.: (x^10+6*x^9+6*x^8+9*x^7+12*x^6+16*x^5+12*x^4+9*x^3+6*x^2+6*x+1) / (x^10-2*x^5+1)
Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1  s11=s11
  Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1  s11=s11
Denom. factorized: (x-1)^2*(x^4+x^3+x^2+x+1)^2
  Denom. factorized: (x^5-1)^2
The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:
The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:
# The coefficient lists are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.
# The coefficient lists for the denominators are always, those for the numerators are often '''symmetrical''' (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.
# The denominator coefficient lists are always symmetrical.  
# All coefficient lists start and end with coefficent 1.
# If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences):
# If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences).
    16 s3 =s3
    42 s5 =s5
    147 s7 =s7
    241 s9 =s9
    539 s11=s11
    589 s13=s13
    199 s15=s15
    114 s17=s17
    160 s19=s19
    128 s21=s21
    110 s23=s23
    96 s25=s25
    82 s27=s27
    85 s29=s29
    24 s31=s31
    16 s33=s33
    33 s35=s35
      4 s37=s37
===Factors of the denominator polynomials ===
===Factors of the denominator polynomials ===
When the denominator polynomials are factorized, there are only 34 different factors (the leading numbers give the occurrences of the factor):
For the denominators of the g.f.s several factorization methods (Mathematica's ''FactorList'' and ''Simplify'', Maple's ''factor'') were tried with different results. In the end, a Maple procedure of Robert Israel was used:
   6536 x-1
myfactor:= proc(p) local X, k, P, T, q;
   2208 x+1
  P:= p; T:= 1;
  2286 x^2+1                              -> y+1
  X:= indets(p)[1];
   1635 x^2-x+1
  k:= degree(P,X);
  3145 x^2+x+1
   while k > 0 do
  1320 x^4+1                              -> y+1
    if rem(P, X^k-1, X, 'q') = 0 then
   1061 x^4-x^2+1                         -> y^2-y+1
        P:= q; T:= T*(X^k-1)
  1246 x^4-x^3+x^2-x+1
    else
  2285 x^4+x^3+x^2+x+1
        k:= k-1
     120 x^6-x^3+1                          -> y^2-y+1
    fi
   1448 x^6+x^3+1                          -> y^2+y+1
  od;
     271 x^6-x^5+x^4-x^3+x^2-x+1
   T * factor(P)
  1474 x^6+x^5+x^4+x^3+x^2+x+1
end proc;
     159 x^8+1                             -> y+1
With this factorization:
     23 x^8-x^4+1                          -> y^2-y+1
* Most of the factors have the form ''x^k-1''.
     54 x^8-x^6+x^4-x^2+1                  -> y^4-y^3+y^2-y+1
* There are no coefficients > 1.
     328 x^8-x^7   +x^5-x^4+x^3   -x+1
* Some of the polynomials could be further reduced by a substitutions ''x^m -> y'' .
      6 x^8+x^7   -x^5-x^4-x^3   +x+1
* After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one polynomial of degree 8 and two polynomials of degree 12.
There are the following 55 different factors in the denominators (with their number of occurrences prefixed):
    892 x+1
  2091 x-1
   1613 x^2+1
    946 x^2+x+1
    233 x^2-1
   1188 x^2-x+1
    981 x^3-1
    975 x^4+1
    621 x^4+x^3+x^2+x+1
    205 x^4-1
    884 x^4-x^2+1
     783 x^4-x^3+x^2-x+1
   1260 x^5-1
    445 x^6+x^3+1
     205 x^6+x^5+x^4+x^3+x^2+x+1
    360 x^6-1
    57 x^6-x^3+1
    122 x^6-x^5+x^4-x^3+x^2-x+1
  1255 x^7-1
     130 x^8+1
    468 x^8-1
     20 x^8-x^4+1
     38 x^8-x^6+x^4-x^2+1
     124 x^8-x^7+x^5-x^4+x^3-x+1
  1144 x^9-1
    145 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
    628 x^10-1
       6 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
       6 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
   1492 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
   1492 x^11-1
    11 x^12-x^6+1                       -> y^2-y+1
      6 x^12+x^11-x^9-x^8+x^6-x^4-x^3+x+1
    34 x^12-x^10+x^8-x^6+x^4-x^2+1      -> y^6-y^5+y^4-y^3+y^2-y+1
    182 x^12-1
      6 x^12+x^11    -x^9-x^8    +x^6    -x^4-x^3    +x+1
      2 x^12-x^10+x^8-x^6+x^4-x^2+1
       5 x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
       5 x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
     32 x^12-x^11     +x^9-x^8   +x^6   -x^4+x^3   -x+1
     12 x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1
    452 x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
      5 x^12-x^6+1
       6 x^16+1                            -> y+1
    452 x^13-1
      6 x^16-x^12+x^8-x^4+1              -> y^4-y^3+y^2-y+1
    117 x^14-1
     89 x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
    198 x^15-1
     12 x^18+x^9+1                        -> y^2+y+1
       6 x^16+1
     59 x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
    29 x^16-1
     18 x^20+x^15+x^10+x^5+1              -> y^4-y^3+y^2-y+1
     89 x^17-1
       6 x^22+x^21+x^20+x^19+x^18+x^17+x^16+x^15+x^14+x^13+x^12+x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
      6 x^18+x^9+1
       6 x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1
     57 x^18-1
We observe the following properties of for the denominators:
     59 x^19-1
# All denominators have a factor ''(x-1)^2''.
    10 x^20-1
# All coefficents of ''x'' in the factors are +1 or -1.
     20 x^21-1
# Except for the factors ''x-1'' and ''x+1'', all factors are of even degree.
       6 x^23-1
# The number of different patterns could be further reduced (to 20) with substitutions ''x^(k*m) -> y^k'' .
       3 x^24-1
# After such substitutions, the factors have the form ''sum(k=0..n: (+-1)^k * x^k)'', except for one pattern of degree 8 and two patterns of degree 12.
    18 x^25-1
      6 x^27-1
    32 x^28-1
      6 x^30-1
      6 x^35-1
      6 x^36-1
      6 x^40-1
===See also===
* [[OEIS/Tilings|Literature and new format]]

Latest revision as of 21:36, 8 September 2023

Generating functions for coordination sequences of uniform tilings

For the 1248 k-uniform tilings determined by Brian Galebach, generating functions (g.f.s) for all corresponding 6536 coordination sequences (coseq.s) were "guessed" by Maple's gfun:guessgf from 128 initial terms. There is a tab-separated file with the result.

As an example we examine the two lines (coseq.s) for Gal.2.13 in this file:

2-Uniform Tiling 13 of 20 Standard notation: [3^6; 3^2.4.12]
A301692 Gal.2.13.1
  O.g.f.: (-x^13-2*x^12+3*x^10+5*x^9+8*x^8+11*x^7+13*x^6+14*x^5+12*x^4+9*x^3+7*x^2+4*x+1) / (x^10-2*x^5+1)
  Coefficients: 1,4,7,9,12,14,13,11,8,5,3,0,-2,-1 / 1,0,0,0,0,-2,0,0,0,0,1  a14>s11
  Denom. factorized: (x^5-1)^2
A301690 Gal.2.13.2 
  O.g.f.: (x^10+6*x^9+6*x^8+9*x^7+12*x^6+16*x^5+12*x^4+9*x^3+6*x^2+6*x+1) / (x^10-2*x^5+1)
  Coefficients: 1,6,6,9,12,16,12,9,6,6,1 / 1,0,0,0,0,-2,0,0,0,0,1  s11=s11
  Denom. factorized: (x^5-1)^2

The resulting rational g.f.s (with a numerator and a denominator polynomial) have several obvious properties:

  1. The coefficient lists for the denominators are always, those for the numerators are often symmetrical (the first half is mirrored, maybe around a center term). This is indicated by "s", or "a" for asymmetrical.
  2. All coefficient lists start and end with coefficent 1.
  3. If both the numerator and the denominator coefficient lists are symmetrical, the degrees are both odd (the leading numbers give the occurrences).

Factors of the denominator polynomials

For the denominators of the g.f.s several factorization methods (Mathematica's FactorList and Simplify, Maple's factor) were tried with different results. In the end, a Maple procedure of Robert Israel was used:

myfactor:= proc(p) local X, k, P, T, q;
  P:= p; T:= 1;
  X:= indets(p)[1];
  k:= degree(P,X);
  while k > 0 do
    if rem(P, X^k-1, X, 'q') = 0 then
       P:= q; T:= T*(X^k-1)
    else
       k:= k-1
    fi
  od;
  T * factor(P)
end proc;

With this factorization:

  • Most of the factors have the form x^k-1.
  • There are no coefficients > 1.
  • Some of the polynomials could be further reduced by a substitutions x^m -> y .
  • After such substitutions, the factors have the form sum(k=0..n: (+-1)^k * x^k), except for one polynomial of degree 8 and two polynomials of degree 12.

There are the following 55 different factors in the denominators (with their number of occurrences prefixed):

   892 x+1
  2091 x-1
  1613 x^2+1
   946 x^2+x+1
   233 x^2-1
  1188 x^2-x+1
   981 x^3-1
   975 x^4+1
   621 x^4+x^3+x^2+x+1
   205 x^4-1
   884 x^4-x^2+1
   783 x^4-x^3+x^2-x+1
  1260 x^5-1
   445 x^6+x^3+1
   205 x^6+x^5+x^4+x^3+x^2+x+1
   360 x^6-1
    57 x^6-x^3+1
   122 x^6-x^5+x^4-x^3+x^2-x+1
  1255 x^7-1
   130 x^8+1
   468 x^8-1
    20 x^8-x^4+1
    38 x^8-x^6+x^4-x^2+1
   124 x^8-x^7+x^5-x^4+x^3-x+1
  1144 x^9-1
   145 x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1
   628 x^10-1
     6 x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
  1492 x^11-1
     6 x^12+x^11-x^9-x^8+x^6-x^4-x^3+x+1
   182 x^12-1
     2 x^12-x^10+x^8-x^6+x^4-x^2+1
     5 x^12-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1
    12 x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1
     5 x^12-x^6+1
   452 x^13-1
   117 x^14-1
   198 x^15-1
     6 x^16+1
    29 x^16-1
    89 x^17-1
     6 x^18+x^9+1
    57 x^18-1
    59 x^19-1
    10 x^20-1
    20 x^21-1
     6 x^23-1
     3 x^24-1
    18 x^25-1
     6 x^27-1
    32 x^28-1
     6 x^30-1
     6 x^35-1
     6 x^36-1
     6 x^40-1

See also