# Seal (discrete mathematics)/stuff

## Sequences and triangles

Each seal can be seen as a set of binary numbers. (Compare the list.)
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 . (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 ).
In these smallest numbers are shown - not ordered by size.

Similarly denotes each seal clan by the smallest number in the clan (i.e. as a value of ). 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 are permutations that assign complementary seal clans to each other - using their index numbers in .
The complements are also shown here for arities up to 7.

shows the number of runs of ones in the binary string of each entry of the Wolka sequence.
The triangle shows that all numbers in ${\displaystyle \{1,...,2^{n-1}\}}$ appear as numbers of runs of ones of n-ary seals, and that the numbers greater than ${\displaystyle 2^{n-2}}$ 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

## 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 N(4,1)1 N(3,1)4 N(3,2)6 N(3,3)4 N(3,4)1 N(2,1)6 N(2,2)12 N(2,3)4 N(2,4)4 N(2,5)3 N(2,6)6 N(1,1)4 N(1,2)6 N(1,3)4 N(1,4)1 N(0)1

(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 .

 All (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 (4) = 67 functions that correspond to entries of 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 (6) = 68 equivalence classes:

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

64 6 1 1 32 5 63 6 16 4 651 16 8 3 1395 22 4 2 651 16 2 1 1 6 15 20 15 6 1 15 60 60 30 6 20 45 90 60 60 15 15 90 10 60 15 20 90 60 15 60 90 180 60 60 90 15 60 60 20 20 90 15 90 45 45 180 30 15 60 20 60 45 90 30 60 60 90 6 15 15 10 60 15 6 15 20 15 6 1 1

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 6506 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 6506.