Seal (discrete mathematics)/stuff

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Sequences and triangles

Each seal can be seen as a set of binary numbers. (Compare the list.)
Template:Oeis shows these numbers ordered by size:

  • 1, (1 until here for Z20)
  • 3, (2 until here for Z21)
  • 5, 9, 15, (5 until here for Z22)
  • 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, (16 until here for Z23)
  • 257, 513, 771, 1025, 1285, 1545, 2049, 2313, 2565, 3075, 3855, 4097, 4369, 4641, 5185, 6273, 8193, 8481, 8721, 9345, 10305, 12291, 13107, 15555, 16385, 16705, 17025, 17425, 18465, 20485, 21845, 23205, 24585, 26265, 26985, 32769, 33153, 33345, 33825, 34833, 36873, 38505, 39321, 40965, 42405, 43605, 49155, 50115, 52275, 61455, 65535, (67 until here for Z24)
  • 65537, ... (continues 374, 2825, 29212...)

The index numbers of this sequence are used to denote the seals - e.g. in the Hasse diagrams shown below.


Each equivalence class has a weight partition, denoted by an index number of Template:Oeis. (Compare this triangle.)



Usually seals (or sona-secs) are denoted by the unique odd number in the sec,
but they could just as well be denoted by the smallest number in the sec (i.e. as a value of Template:Oeis).
In Template:Oeis these smallest numbers are shown - not ordered by size.

Similarly Template:Oeis denotes each seal clan by the smallest number in the clan (i.e. as a value of Template:Oeis). This sequence is ordered by size:

  • 1, (1 until here for Z20)
  • 3, (2 until here for Z21)
  • 6, 15, (4 until here for Z22)
  • 24, 60, 105, 255, (8 until here for Z23)
  • 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, (16 until here for Z24)
  • 98304, ... (continues 32, 68, 148...)


The rows of the triangle Template:Oeis are permutations that assign complementary seal clans to each other - using their index numbers in Template:Oeis.
The complements are also shown here for arities up to 7.

Template:Oeis shows the number of runs of ones in the binary string of each entry of the Wolka sequence.
The triangle Template:Oeis shows that all numbers in {1,...,2n−1} appear as numbers of runs of ones of n-ary seals, and that the numbers greater than 2n−2 appear exactly twice.
E.g. the numbers of runs of ones of 4-ary seals range from 1 to 8, and those greater 4 appear exactly twice. (The two 4-ary seals with 8 runs of ones are 46 and 61 - compare the list.)

Z22

w:Klein four-group


Z23



Z24




The number of elements by rank in the lattice is 1, 15, 35, 15, 1.
This is row 4 in the triangle of 2-binomial coefficients.

Each subgroup of Z24 corresponds to a 4-ary Boolean function, and thus could be represented by a binarily colored tesseract.
One may ask, whether two such tesseracts are essentially the same or not,
i.e. if one can be turned into the other by any rotation of the tesseract.
But the answer is less easy than for the 3-dimensional case.

All binarily colored tesseracts that can be turned into each other form a big equivalence class.
The files linked in the following table list the corresponding 4-ary Boolean functions in the big equivalence classes.

16 4 N(4,1)1 1 1
8 3 N(3,1)4 N(3,2)6 N(3,3)4 N(3,4)1 15 4
4 2 N(2,1)6 N(2,2)12 N(2,3)4 N(2,4)4 N(2,5)3 N(2,6)6 35 6
2 1 N(1,1)4 N(1,2)6 N(1,3)4 N(1,4)1 15 4
1 0 N(0)1 1 1
Order
of subgroups 		Rank
in lattice 		Equivalence classes (becs)
(Subscripts show the number of subgroups in the equivalence class.) 		Number of
subgroups 		Number of
e. c.

(The table is organized upside down, so the rows are arranged like the layers in an Hasse diagram.)

There are 16 of 402 equivalence classes. Their number by rank in the lattice is 1, 4, 6, 4, 1.
This is row 4 in Template:Oeis.


All Template:Oeis(4) = 307 functions in these equivalence classes (in a matrix like this one)
The respective matrices for smaller sona are the 16x16, 4x4, 2x2 and 1x2 submatrices.
All Template:Oeis(4) = 67 functions that correspond to entries of Template:Oeis
These are the entries in the odd columns of the matrix on the left.


Z25

Z25 has 1 + 31 + 155 + 155 + 31 + 1 = 374 subgroups of order 1, 2, 4, 8, 16, 32.

The number of equivalence classes is 32.

Equivalence classes belonging together as counterparts: 

N(0)       N(5,01)
       
N(1,01)    N(4,01)
N(1,02)    N(4,02)
N(1,03)    N(4,03)
N(1,04)    N(4,04)
N(1,05)    N(4,05)
       
N(2,01)    N(3,01)
N(2,02)    N(3,02)
N(2,03)    N(3,05)
N(2,04)    N(3,03)
N(2,05)    N(3,06)
N(2,06)    N(3,07)
N(2,07)    N(3,04)
N(2,08)    N(3,08)
N(2,09)    N(3,09)
N(2,10)    N(3,10)

Z26

This text refers to the table of all 2825 subgroups: Subgroups of Z2^6     (very large)

Z26 has 1 + 63 + 651 + 1395 + 651 + 63 + 1 = 2825 subgroups of order 1, 2, 4, 8, 16, 32, 64.

These are the Template:Oeis(6) = 68 equivalence classes:

(m,n) with 0≤m≤6 and 1≤n≤Template:Oeis(6,m)
The table entries show the number of subgroups in the equivalence class (m,n).

64 6 1 1 1
32 5 6 15 20 15 6 1 63 6
16 4 15 60 60 30 6 20 45 90 60 60 15 15 90 10 60 15 651 16
8 3  20  90  60  15  60  90 180  60  60  90  15  60  60  20  20  90  15  90  45  45 180  30 1395 22
4 2 15 60 20 60 45 90 30 60 60 90 6 15 15 10 60 15 651 16
2 1 6 15 20 15 6 1 63 6
1 0 1 1 1
Order
of subgroups		 Rank
in lattice		 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Number of
subgroups		 Number of
e. c.
The elements of four equivalence classes:
N(2,11) and N(4,5) with 6 seals each are counterparts.
N(1,6) and N(5,6) with one seal each are counterparts.

In the big table it can be seen that the binary string's Walsh spectra share the pattern of another binary string.
So each seal has a counterpart with the index number shown in column ♭.
These pairs of counterpart seals belong to pairs of counterpart equivalence classes.
10 equivalence classes N(3,k) are their own counterparts. In one of them - N(3,17) - even the seals themselves are their own counterparts.

N(0) N(6,01) 1
N(1,06) N(5,06) 1
N(1,01) N(5,01) 6
N(1,05) N(5,05) 6
N(1,02) N(5,02) 15
N(1,04) N(5,04) 15
N(1,03) N(5,03) 20
N(2,11) N(4,05) 6
N(2,14) N(4,14) 10
N(2,01) N(4,01) 15
N(2,12) N(4,11) 15
N(2,16) N(4,16) 15
N(2,13) N(4,12) 15
N(2,03) N(4,06) 20
N(2,07) N(4,04) 30
N(2,05) N(4,07) 45
N(2,02) N(4,02) 60
N(2,04) N(4,03) 60
N(2,08) N(4,09) 60
N(2,09) N(4,10) 60
N(2,15) N(4,15) 60
N(2,06) N(4,08) 90
N(2,10) N(4,13) 90
N(3,04) N(3,11) 15
N(3,17) 15
N(3,01) 20
N(3,14) 20
N(3,15) 20
N(3,22) 30
N(3,19) N(3,20) 45
N(3,03) N(3,05) 60
N(3,08) N(3,12) 60
N(3,09) N(3,13) 60
N(3,02) 90
N(3,06) 90
N(3,10) N(3,16) 90
N(3,18) 90
N(3,07) 180
N(3,21) 180


Each equivalence class has a weight partition, but some have the same. Weight partition Template:Font is the first one that does not identify an equivalence class. It belongs both to N(3,15) and N(3,17).


The 20 elements of N(3,15)
The 15 elements of N(3,17)
N(3,15) and N(3,17) have the same weight partition Template:Font.
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