Enumeration of classes

Popis: D:\binary\images\d_gclsf.jpg





Let us evaluate the totals of classes v(n,k),w(n,k) and m(n,k) for particular orders k as functions of n:

Self classes v(n,k)

    k  fv(n)
    1  1/1 (n) = n
    2  1/2 (n^2 -n)
    3  1/3 (n^3 -n)
    4  1/4 (n^4-(n^2-n)-(n)) = 1/4 (n^4 -n^2)
    5  1/5 (n^5 -n)
    6  1/6 (n^6-(n^3-n)-(n^2-n)-(n))= 1/6 (n^6 -n^3-n^2+ n)
    7  1/7 (n^7 -n)
    8  1/8 (n^8-(n^4-n^2)-(n^2-n)-(n))=1/8 (n^8 -n^4)
    9  1/9 (n^9 -n^3)
   10  1/10(n^10-n^5-n^2+n)
   11  1/11(n^11-n)
   12  1/12(n^12-n^6-n^4+n^2)
   13  1/13(n^13-n)
  k\n    1  2   3    4     5      6      7       8
    1    1  2   3    4     5      6      7       8
    2    0  1   3    6    10     15     21      28
    3    0  2   8   20    40     70    112     168
    4    0  3  18   60   150    315    588    1008
    5    0  6  48  204   624   1554   3360    6552
    6    0  9 116  670  2580   7735  19544   43596
    7    0 18 312 2340 11160  39990 117648  299592
    8    0 30 810 8160 48750 209790 720300 2096640

Nested classes w(n,k)

  k  fw(n)
  1  0
  2  n
  3  n
  4  1/2(n^2-n)+1/1(n)= 1/2 (n^2+ n)
  5  n
  6  1/3(n^3-n)+1/2(n^2-n)+1/1(n)= 1/6 (2n^3+3n^2+n)
  7  n
  8  1/4(n^4-n^2)+1/2(n^2-n)+n = 1/4 (n^4+n^2+2n)
  k\n    1  2   3    4     5     6     7    8
    1    0  0   0    0     5     6     7    8
    2    1  2   3    4     5     6     7    8
    3    1  2   3    4     5     6     7    8
    4    1  3   6   10    15    21    28   36
    5    1  2   3    4     5     6     7    8
    6    1  5  14   30    55    91   140  204
    7    1  2   3    4     5     6     7    8
    8    1  6  24   70   165   336   616 1044

All classes m(n,k)

  k  fm(n)
  1  1/1 (n) + 0 = n
  2  1/2 (n^2- n)+ n = 1/2 (n^2+n)
  3  1/3 (n^3-n)+ n = 1/3 (n^3+ 2n)
  4  1/4 (n^4-n2)+ 1/2 (n^2+n) = 1/4 (n^4+ n^2+2n)
  5  1/5 (n^5-n) + n = 1/5 (n^5+ 4n)
  6  1/6 (n^6-n^3-n^2+n)+1/6(2n^3+3n^2+n)  = 1/6 (n^6+n^3+2n^2+2n)
  7  1/7 (n^7-n) + n = 1/7 (n^7+ 6n)
  8  1/8 (n^8-n^4)+ 1/4(n^4+n^2+2n)  = 1/8 (n^8+n^4+2n^2+4n)
    k\n  1  2   3    4     5      6      7       8
    1    1  2   3    4     5      6      7       8
    2    1  3   6   10    15     21     28      36
    3    1  4  11   24    45     76    119     176
    4    1  6  24   70   165    336    616    1044
    5    1  8  51  208   629   1560   3367    6560
    6    1 14 130  700  2635   7826  19684   43800
    7    1 20 315 2344 11165  39996 117655  299600
    8    1 36 834 8230 48915 210126 720916  209768

Outline G(2,k)

       k    v(2,k)   w(2,k) m(2,k)
     1      2    0      2
     2      1    2      3
     3      2    2      4
     4      3    3      6
     5      6    2      8
     6      9    5     14
     7     18    2     20
     8     30    6     36
     9     56    4     60
    10     99    9    108
    11    186    2    188
    12    335   17    352

Segmentation

  k       fv(n+1)-fv(n)
  1   1
  2   n*1
  3   n*(n+1)
  4   (n/2)*(2n^2+3n+1)
  5   n*(n^3+2n^2+2n+1)
  6   (n/6)*(6n^4+15n^3+20n^2+12n+1)
  7   n*(n^5+3n^4+5n^3+5n^2+3n+1)
  8   (n/4)*(4n^6+14n^5+28n^4+35n^3+26n^2+11n+4)
    k\n  1  2   3    4     5      6      7       8
    1    1  1   1    1     1      1      1       1
    2    0  1   2    3     4      5      6       7
    3    0  2   6   12    20     30     42      56
    4    0  3  15   42    90    165    270     423
    5    0  6  42  156   420    930   1806    3192
    6    0  9 107  554  1910   5155  11809   24052
    7    0 18 294 2028  8820  28830  77658  181944
    8    0 30 780 7350 40590 161040 510510 1376340

In case k=4: 3= 1+2; 15= 4+5+6; 42= 9+10+11+12; 90= 16+17+18+19+20;


Schematic algebra