# Seal (discrete mathematics)

Seal is a neologism for a mathematical object, that is essentially a subgroup of nimber addition. The addition of nimbers is the bitwise XOR of non-negative integers. For a finite set $\{0,...,2^{n}-1\}$ it forms the Boolean group Z$_2$$^n$.
A seal shall be defined as a Boolean function whose family matrix is also the matrix of an equivalence relation. This implies, that the Boolean function is odd (i.e. that the first entry of it's truth table is true), and that it is the unique odd function in its family.
A seal is also a periodic set partition. The members of its family shall be called its blocks.
The weight of a Boolean function is be the quotient of the sum and the length of its truth table. The weight of a seal is ${\frac {1}{2^{d}}}$, where $d$ is its depth. The unique seal with depth 0 is the tautology. The seals with depth 1 are the negated variadic XORs with one or more arguments.

## cccurences

The fixed points of Walsh permutations are seals. The symmetry of Boolean functions is related to seals.

## equivalenc classes
### house

Each seal belongs to a faction, which shall be called its house. A house can also be seen as a clan of seal blocks. So a house can the represented by the smallest Zhegalkin index of the faction or the clan. (That of the faction is easier to calculate.)

## antipode

The seals with arity a form a symmetric Hasse diagram, whose top node is the tautology. From top to bottom the layers are depth = 0...a. From bottom to top they are rank = 0...a. The number of seals in layer n is $\mathrm {Oak} (a,n)$. For the given arity each seal has a Walsh spectrum. It's non-zero entries are $2^{rank}$ (the weight of the seal's finite truth table). Their pattern describes another seal in the opposite layer of the Hasse diagram, which shall be called its antipode. When a seal in house x has an antipode in house y, then all seals in x have antipodes in y. Each house has an antipode for a given arity.
The following seal is the antipode of the example shown above. Their depth (and rank) is 2, and both are on the middle layer of the Hasse diagram.
1001 0000 0000 1001 (Ж 1911) has cohort {0, 3, 12, 15}. So for arity 4 it is its own antipode. (See this file.)

## cohort and legion

While a seal has different antipodes for each arity, they all start with the same binary pattern, and then continue with zeros. In other words, all antipodes of a seal can be described as the same finite set of integers (which corresponds to an entry of sequence Rose). This set of integers shall be called cohort. Every cohort belongs to a legion. Cohort and legion are properties of Boolean functions in general – not just of seals.
The cardinalities of cohort and legion are a powers of two. The exponent for the cohort is depth. The exponent for the legion shall be called gravity. Gravity is similar to valency. The pyramids TwistedLiana and Lonicera have it as one of their dimensions.

## integer representations

truth tables and Zhegalkin indices   (Hasse diagrams of Venn diagrams) :
| | truth tables (entries of Rose) | Zhegalkin indices (entries of Tulip) |
| arity 2 | | |
| arity 3 | | |
sequences Rose and Tulip :
Python program to calculate the integer sequence :

## quantities

overview of sequences, triangles and pyramids :
pairs :
| | | houses | rooms | ? | seals | blocks |
| sequences | a | Clover Heather | Basil MaimedClover | Valerian ValerianDrop | Daisy Dahlia | DahliaCore Azalea |
| triangles | a, d | Elm Lime | Aronia MaimedElm | Juniper JuniperDrop | Oak Maple | MapleCore Sycamore |
| triangles | a, v | | Rowan RowanDrop | Sloe SloeDrop | Ash Aspen | |
| triangles | a, g | | Elder ElderDrop | TurnedSloe TurnedSloeDrop | TwistedAsh Alder | |
| pyramids | a, d, v | | | | Liana Ivy | |
| pyramids | a, d, g | | | | TwistedLiana Lonicera | |
| hyper- pyramids | a, d, v, g | empty | Wisteria WisteriaDrop | Syringa SyringaDrop | Jacaranda JacarandaDrop | |
| a | 🌊 arity 💧 adicity |
| d | depth |
| v | valency |
| g | gravity |
sequences :
| name | OEIS | description | entries |
| Rose | A190939 | truth tables | 1, 3, 5, 9, 15, 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255 |
| Tulip | | Zhegalkin indices | 1, 3, 5, 7, 15, 17, 19, 21, 23, 51, 63, 85, 95, 119, 127, 255 |
| SmallLily | A305737 | | 1, 2, 8, 184, 62464 |
| | | Betula sums | 1, 2, 18, 772, 131238, 82795124, 194513625552, 1734587910632112 |
| | | Betula diagonal | 1, 1, 13, 670, 122861, 80189094, 191467330714, 1721026866650520 |
| Aster | A135922 | | 1, 1, 2, 6, 26, 158, 1330, 15414 |
| AsterDrop | | | 1, 0, 1, 4, 20, 132, 1172, 14084 |
| Daisy | A006116 | | 1, 2, 5, 16, 67, 374, 2825, 29212 |
| Dahlia | ~A182176 | | 1, 1, 3, 11, 51, 307, 2451, 26387 |
| Clover | A076766 | | 1, 2, 4, 8, 16, 32, 68, 148 |
| Heather | ~A034343 | | 1, 1, 2, 4, 8, 16, 36, 80 |
| DahliaCore | A182176 | | 1, 3, 11, 51, 307, 2451, 26387, 387987 |
| Azalea | | | 1, 2, 8, 40, 256, 2144, 23936, 361600 |
| Valerian | | | 1, 2, 5, 16, 61, 264, 1225, 5790 |
| ValerianDrop | | | 1, 1, 3, 11, 45, 203, 961, 4565 |
| Basil | | | 1, 2, 5, 12, 27, 58, 125, 272 |
| MaimedClover | | | 1, 1, 3, 7, 15, 31, 67, 147 |
| Verbena | | | 1, 1, 2, 6, 20, 78, 300, 1092 |
triangles :
| name | OEIS | description |
| Birch | ~A139382 | |
| Betula | A156823 | part of Birch |
| Oak | A022166 | |
| Maple | ~A289537 | |
| Ash | | |
| Aspen | | |
| TwistedAsh | | |
| Alder | | |
| Elm | A076831 | |
| Lime | ~A034253 | |
| Tilia | ~A076832 | Difference of neighbors gives entry of Lime. |
| MapleCore | ~A289537 | |
| Sycamore | | |
| TwistedBirch | ~A139382 | |
| TwistedBirchDrop | | |
seals and houses by arity and depth:   symmetric triangles Oak and Elm :
| 🌊 triangle Oak (A022166) row sums Daisy (A006116) | 🌊 triangle Elm (A076831) row sums Clover (A076766) |
| 0 1 2 3 4 5 6 7 0 1 1 1 1 1 2 2 1 3 1 5 3 1 7 7 1 16 4 1 15 35 15 1 67 5 1 31 155 155 31 1 374 6 1 63 651 1395 651 63 1 2825 7 1 127 2667 11811 11811 2667 127 1 29212 | 0 1 2 3 4 5 6 7 0 1 1 1 1 1 2 2 1 2 1 4 3 1 3 3 1 8 4 1 4 6 4 1 16 5 1 5 10 10 5 1 32 6 1 6 16 22 16 6 1 68 7 1 7 23 43 43 23 7 1 148 |
blocks a × d:   triangles MapleMinor and Sycamore :
🌊    triangle MapleCore (~A289537)    row sums DahliaCore (A182176) :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 2 2 | | | | | | | 3 |
| 2 | 1 · 1 1 | 3 · 2 6 | 1 · 4 4 | | | | | | 11 |
| 3 | 1 · 1 1 | 7 · 2 14 | 7 · 4 28 | 1 · 8 8 | | | | | 51 |
| 4 | 1 · 1 1 | 15 · 2 30 | 35 · 4 140 | 15 · 8 120 | 1 · 16 16 | | | | 307 |
| 5 | 1 · 1 1 | 31 · 2 62 | 155 · 4 620 | 155 · 8 1240 | 31 · 16 496 | 1 · 32 32 | | | 2451 |
| 6 | 1 · 1 1 | 63 · 2 126 | 651 · 4 2604 | 1395 · 8 11160 | 651 · 16 10416 | 63 · 32 2016 | 1 · 64 64 | | 26387 |
| 7 | 1 · 1 1 | 127 · 2 254 | 2667 · 4 10668 | 11811 · 8 94488 | 11811 · 16 188976 | 2667 · 32 85344 | 127 · 64 8128 | 1 · 128 128 | 387987 |
💧    triangle Sycamore    row sums Azalea :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 0 | 1 · 2 2 | | | | | | | 2 |
| 2 | 0 | 2 · 2 4 | 1 · 4 4 | | | | | | 8 |
| 3 | 0 | 4 · 2 8 | 6 · 4 24 | 1 · 8 8 | | | | | 40 |
| 4 | 0 | 8 · 2 16 | 28 · 4 112 | 14 · 8 112 | 1 · 16 16 | | | | 256 |
| 5 | 0 | 16 · 2 32 | 120 · 4 480 | 140 · 8 1120 | 30 · 16 480 | 1 · 32 32 | | | 2144 |
| 6 | 0 | 32 · 2 64 | 496 · 4 1984 | 1240 · 8 9920 | 620 · 16 9920 | 62 · 32 1984 | 1 · 64 64 | | 23936 |
| 7 | 0 | 64 · 2 128 | 2016 · 4 8064 | 10416 · 8 83328 | 11160 · 16 178560 | 2604 · 32 83328 | 126 · 64 8064 | 1 · 128 128 | 361600 |
seals a × d:   triangles Oak and Maple :
🌊    triangle Oak (A022166)    row sums Daisy (A006116) :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 3 | 1 | | | | | | 5 |
| 3 | 1 | 7 | 7 | 1 | | | | | 16 |
| 4 | 1 | 15 | 35 | 15 | 1 | | | | 67 |
| 5 | 1 | 31 | 155 | 155 | 31 | 1 | | | 374 |
| 6 | 1 | 63 | 651 | 1395 | 651 | 63 | 1 | | 2825 |
| 7 | 1 | 127 | 2667 | 11811 | 11811 | 2667 | 127 | 1 | 29212 |
💧    triangle Maple (~A289537)    row sums Dahlia (~A182176) :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 2 | 1 | | | | | | 3 |
| 3 | 0 | 4 | 6 | 1 | | | | | 11 |
| 4 | 0 | 8 | 28 | 14 | 1 | | | | 51 |
| 5 | 0 | 16 | 120 | 140 | 30 | 1 | | | 307 |
| 6 | 0 | 32 | 496 | 1240 | 620 | 62 | 1 | | 2451 |
| 7 | 0 | 64 | 2016 | 10416 | 11160 | 2604 | 126 | 1 | 26387 |
houses a × d:   triangles Elm and Lime :
🌊    triangle Elm (A076831)    row sums Clover (A076766) :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 1 | 1 | | | | | | | 2 |
| 2 | 1 | 2 | 1 | | | | | | 4 |
| 3 | 1 | 3 | 3 | 1 | | | | | 8 |
| 4 | 1 | 4 | 6 | 4 | 1 | | | | 16 |
| 5 | 1 | 5 | 10 | 10 | 5 | 1 | | | 32 |
| 6 | 1 | 6 | 16 | 22 | 16 | 6 | 1 | | 68 |
| 7 | 1 | 7 | 23 | 43 | 43 | 23 | 7 | 1 | 148 |
💧    triangle Lime (~A034253)    row sums Heather (~A034343) :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 | | | | | | | | 1 |
| 1 | 0 | 1 | | | | | | | 1 |
| 2 | 0 | 1 | 1 | | | | | | 2 |
| 3 | 0 | 1 | 2 | 1 | | | | | 4 |
| 4 | 0 | 1 | 3 | 3 | 1 | | | | 8 |
| 5 | 0 | 1 | 4 | 6 | 4 | 1 | | | 16 |
| 6 | 0 | 1 | 6 | 12 | 11 | 5 | 1 | | 36 |
| 7 | 0 | 1 | 7 | 21 | 27 | 17 | 6 | 1 | 80 |
seals a × v:   triangles Ash and Aspen :
🌊    triangle Ash    row sums Daisy (A006116) :
| va | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 1 1 | | | | | | | 2 |
| 2 | 1 · 1 1 | 2 · 1 2 | 1 · 2 2 | | | | | | 5 |
| 3 | 1 · 1 1 | 3 · 1 3 | 3 · 2 6 | 1 · 6 6 | | | | | 16 |
| 4 | 1 · 1 1 | 4 · 1 4 | 6 · 2 12 | 4 · 6 24 | 1 · 26 26 | | | | 67 |
| 5 | 1 · 1 1 | 5 · 1 5 | 10 · 2 20 | 10 · 6 60 | 5 · 26 130 | 1 · 158 158 | | | 374 |
| 6 | 1 · 1 1 | 6 · 1 6 | 15 · 2 30 | 20 · 6 120 | 15 · 26 390 | 6 · 158 948 | 1 · 1330 1330 | | 2825 |
| 7 | 1 · 1 1 | 7 · 1 7 | 21 · 2 42 | 35 · 6 210 | 35 · 26 910 | 21 · 158 3318 | 7 · 1330 9310 | 1 · 15414 15414 | 29212 |
💧    triangle Aspen    row sums Dahlia (~A182176) :
| va | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 0 | 1 · 1 1 | | | | | | | 1 |
| 2 | 0 | 1 · 1 1 | 1 · 2 2 | | | | | | 3 |
| 3 | 0 | 1 · 1 1 | 2 · 2 4 | 1 · 6 6 | | | | | 11 |
| 4 | 0 | 1 · 1 1 | 3 · 2 6 | 3 · 6 18 | 1 · 26 26 | | | | 51 |
| 5 | 0 | 1 · 1 1 | 4 · 2 8 | 6 · 6 36 | 4 · 26 104 | 1 · 158 158 | | | 307 |
| 6 | 0 | 1 · 1 1 | 5 · 2 10 | 10 · 6 60 | 10 · 26 260 | 5 · 158 790 | 1 · 1330 1330 | | 2451 |
| 7 | 0 | 1 · 1 1 | 6 · 2 12 | 15 · 6 90 | 20 · 26 520 | 15 · 158 2370 | 6 · 1330 7980 | 1 · 15414 15414 | 26387 |
illustrated examples:       Oak, Maple (seals with arity 5)       MapleMinor, Sycamore (blocks with adicity 4) :
triangles :

### number pyramids

See pyramid Liana. These pyramid pairs are refinements of Oak and Maple, adding axes for valency and gravity.