Seal (discrete mathematics)

Studies of Boolean functions

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 Z₂ⁿ.

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.

seal 1001 0000 0110 0000   (Ж 4471)
This seal has adicity 4, and is shown with (the lowest possible) arity 4, i.e. with a truth table of length 16. The 16×16 matrix on the left is its family matrix. It also describes a partition of the set $\{0\ldots 15\}$ into four blocks, which are shown in different colors. On the right the set partition is shown as a vertex coloring of the tesseract graph.   (The 4×4 matrix shows essentially the same.)

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.

negated binary Walsh matrix
The seals with depth 1 are the positive rows of a negated binary Walsh matrix.   (The tautology with depth 0 is in the top row.)

Occurences

The fixed points of Walsh permutations are seals.
The symmetry of any Boolean function is the symmetry of a seal.

Equivalence classes

Equivalence classes (EC) of seals are factions. Those of blocks are clans.
Thus an EC can the represented by the smallest Zhegalkin index of the faction or the clan.   (That of the faction is easier to calculate.)

Antipodes

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 EC x has an antipode in EC y, then all seals in x have antipodes in y. Each EC has an antipode for a given arity. (Some EC are their own antipodes.)

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.

seal 1000 1000 0001 0001   (Ж 1807)
The Walsh spectrum for arity 4 is (4, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0). Its non-zero entries form the pattern 1001 0000 0110 0000 of the antipode Ж 4471.

Cohorts

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 with the same finite set of integers (which corresponds to an entry of sequence Rose).
This set of integers shall be called cohort. It is a property of Boolean functions in general – not just of seals.

Integer representations

sequences Rose and Tulip

Like Boolean functions in general, seals should be seen as periodic truth tables (corresponding to fractions). For a given arity they can be seen as finite truth tables (corresponding to integers). But this approach shall be avoided here. The integer values of the finite truth tables form the sequence Rose = 1, 3, 5, 9, 15... (A190939)     An illustration can be seen here. sequence Rose for arity 4 (adicities 0...4) 1, 3, 5, 9, 15, 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, 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 Hasse diagrams for arities 2 and 3 The gray numbers are the keys to sequence Rose. This is a consistent numbering only for a given arity, but not across arities. Seals with the same label are not equal. E.g. label 0 stands for $\neg A\land \neg B$ on the left, but for $\neg A\land \neg B\land \neg C$ on the right. Equal seals do not have the same label. E.g. the tautology is labeled 4 on the left, but 15 on the right. arity 2 arity 3 Python program to calculate the integer sequence This code uses the Python library discrete helpers. from discretehelpers.a import make_linear_binv arity = 4 depth_to_seals = {depth: None for depth in range(arity)} depth_to_seals[0] = [make_linear_binv(walsh=0, parity=1, arity=arity)] # tautology depth_to_seals[1] = [make_linear_binv(walsh=i, parity=1, arity=arity) for i in range(1, 2 ** arity)] # negated Walsh functions for new_depth in range(2, arity+1): new_weight = 2 ** (arity - new_depth) old_depth = new_depth - 1 old_seals = depth_to_seals[old_depth] new_seals = set() for key_a, seal_a in enumerate(old_seals): for key_b in range(key_a + 1, len(old_seals)): seal_b = old_seals[key_b] candidate = seal_a & seal_b if candidate.weight == new_weight: new_seals.add(candidate) depth_to_seals[new_depth] = sorted(new_seals) lengths = [] sequence = [] for depth, seals in depth_to_seals.items(): rank = arity - depth seal_integers = sorted([seal.intval for seal in seals]) print(f'{rank}: {seal_integers},') lengths.append(len(seal_integers)) sequence += seal_integers print('\n', lengths) print('\n', sorted(sequence)) resulting dictionary rank_to_seals = { 4: [65535], 3: [255, 3855, 13107, 15555, 21845, 23205, 26265, 26985, 38505, 39321, 42405, 43605, 50115, 52275, 61455], 2: [15, 51, 85, 105, 153, 165, 195, 771, 1285, 1545, 2313, 2565, 3075, 4369, 4641, 5185, 6273, 8481, 8721, 9345, 10305, 12291, 16705, 17025, 17425, 18465, 20485, 24585, 33153, 33345, 33825, 34833, 36873, 40965, 49155], 1: [3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769], 0: [1] } A better way to represent the seals by integers is with Zhegalkin indices. They form the sequence Tulip = 1, 3, 5, 7, 15... sequence Tulip for arity 4 (adicities 0...4) 1, 3, 5, 7, 15, 17, 19, 21, 23, 51, 63, 85, 95, 119, 127, 255, 257, 259, 261, 263, 273, 275, 277, 279, 771, 783, 819, 831, 1285, 1295, 1365, 1375, 1799, 1807, 1911, 1919, 3855, 4095, 4369, 4403, 4437, 4471, 4883, 4915, 4959, 4991, 5397, 5439, 5461, 5503, 5911, 5951, 5983, 6007, 13107, 13311, 16191, 16383, 21845, 22015, 24415, 24575, 30583, 30719, 32639, 32767, 65535 illustration:   Zhegalkin indices and truth tables for arity 4 The rows of the green matrix are the twins of those in the red matrix. The latter are the periodic truth tables of the seals. The green integers are their Zhegalkin indices. The tables are divided into adicities 0...4, corresponding to the period length (which is indicated by the grayed-out area). The depth is indicated by the numbers between the two tables.

Integer sequences

table of sequences, triangles and pyramids

a 🌊 arity 💧 adicity d depth v valency p pennon weight non-counting sequence sequence triangle pyramid Arity n is short for adicity ≤ n. Sequences, triangles and pyramids using arity are marked with a wave 🌊, those using adicity are marked with a drop 💧. 🌊 entries are sums of 💧 columns. </noinclude> </noinclude> name OEIS description 🌊 ↔ 💧 children entries 1 Rose A190939 truth tables 1, 3, 5, 9, 15, 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255 1 Tulip Zhegalkin indices 1, 3, 5, 7, 15, 17, 19, 21, 23, 51, 63, 85, 95, 119, 127, 255 3 💧 Ivy (a, d, v) ↦ # seals Liana layer sums Maple and Aspen (with row sums Dahlia) side Birch 2 💧 Maple ~A289537 (a, d) ↦ # seals Oak row sums Dahlia 2 💧 Aspen (a, v) ↦ # seals Ash row sums Dahlia diagonal Aster 1 💧 Dahlia ~A182176 a ↦ # seals Daisy 1, 1, 3, 11, 51, 307, 2451, 26387 3 🌊 Liana (a, d, v) ↦ # seals Ivy layer sums Oak and Ash (with row sums Daisy) side Birch 2 🌊 Oak A022166 (a, d) ↦ # seals symmetric triangle Maple row sums Daisy 2 🌊 Ash (a, v) ↦ # seals Aspen row sums Daisy diagonal Aster 1 🌊 Daisy A006116 a ↦ # seals Dahlia 1, 2, 5, 16, 67, 374, 2825, 29212 2 💧 Lime A034253 (a, d) ↦ # EC Elm row sums Heather 1 💧 Heather ~A034343 a ↦ # EC Clover 1, 1, 2, 4, 8, 16, 36, 80 2 🌊 Elm A076831 (a, d) ↦ # EC symmetric triangle Lime row sums Clover 1 🌊 Clover A076766 a ↦ # EC Heather 1, 2, 4, 8, 16, 32, 68, 148 2 Birch ~A139382 (a, d) ↦ # new seals row sums Aster 1 Aster A135922 a ↦ # new seals inverse binomial transform of Daisy 1, 1, 2, 6, 26, 158, 1330, 15414 2 🌊 MapleMinor ~A289537 (a, d) ↦ # blocks Sycamore row sums DahliaMinor 1 🌊 DahliaMinor A182176 a ↦ # blocks Azalea 1, 3, 11, 51, 307, 2451, 26387, 387987 2 💧 Sycamore (a, d) ↦ # blocks MapleMinor row sums Azalea 1 💧 Azalea a ↦ # blocks DahliaMinor 1, 2, 8, 40, 256, 2144, 23936, 361600 2 Beech ~A076832 ? Difference of neighbors gives entry of Lime. row sums Poppy diagonal Heather 1 Poppy ? 1, 1, 3, 8, 20, 48, 128, 326 2 💧 Alder (a, p) ↦ # seals reflected Ash row sums Dahlia 1 SmallMeshima A305737 ?    Meshima − 1 1, 2, 8, 184, 62464

Quantities

seals and EC by arity and depth:   symmetric triangles Oak and Elm

The triangle Oak (left) shows the number of seals by arity (rows) and depth (columns). These are 2-binomial coefficients.   Arity n is short for adicity ≤ n. The triangle Elm (right) shows the corresponding numbers of equivalence classes.   (Not to be confused with Pascal's triangle.) 🌊    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 E.g. there are $\mathrm {Oak} (4,2)=35$ 4-ary seals of depth 2, and they belong to $\mathrm {Elm} (4,2)=6$ different equivalence classes.

blocks a × d:   triangles MapleMinor and Sycamore

Oak with columns multiplied by powers of two. 🌊    triangle MapleMinor (~A289537)    row sums DahliaMinor (A182176) d a 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 Maple with columns multiplied by powers of two.    (compare) 💧    triangle Sycamore    row sums Azalea d a 0 1 2 3 4 5 6 7 sums 0 1 · 1 1 1 1 0 · 1 0 1 · 2 2 2 2 0 · 1 0 2 · 2 4 1 · 4 4 8 3 0 · 1 0 4 · 2 8 6 · 4 24 1 · 8 8 40 4 0 · 1 0 8 · 2 16 28 · 4 112 14 · 8 112 1 · 16 16 256 5 0 · 1 0 16 · 2 32 120 · 4 480 140 · 8 1120 30 · 16 480 1 · 32 32 2144 6 0 · 1 0 32 · 2 64 496 · 4 1984 1240 · 8 9920 620 · 16 9920 62 · 32 1984 1 · 64 64 23936 7 0 · 1 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) d a 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 Offset Oak with diagonals multiplied by powers of two. 💧    triangle Maple (~A289537)    row sums Dahlia (~A182176) d a 0 1 2 3 4 5 6 7 sums 0   1 1 1   0 1 · 1 1 1 2   0 1 · 2 2 1 · 1 1 3 3   0 1 · 4 4 3 · 2 6 1 · 1 1 11 4   0 1 · 8 8 7 · 4 28 7 · 2 14 1 · 1 1 51 5   0 1 · 16 16 15 · 8 120 35 · 4 140 15 · 2 30 1 · 1 1 307 6   0 1 · 32 32 31 · 16 496 155 · 8 1240 155 · 4 620 31 · 2 62 1 · 1 1 2451 7   0 1 · 64 64 63 · 32 2016 651 · 16 10416 1395 · 8 11160 651 · 4 2604 63 · 2 126 1 · 1 1 26387

EC a × d:   triangles Elm and Lime
🌊    triangle Elm (A076831)    row sums Clover (A076766) d a 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) d a 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

Pascal's triangle with columns multiplied by Aster. 🌊    triangle Ash    row sums Daisy (A006116) v a 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 Offset Pascal's triangle with columns multiplied by Aster. 💧    triangle Aspen    row sums Dahlia (~A182176) v a 0 1 2 3 4 5 6 7 sums 0   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

triangle Birch with row sums Aster
These sequences count seals whose adicity and valency are equal. $\mathrm {Aster} (n)$ is the number of seals whose adicity and valency is n. Among them $\mathrm {Birch} (n,d)$ are those with depth d.
triangle Birch (~A139382)    row sums Aster (A135922) d n 0 1 2 3 4 5 6 7 sums 0 1 1 1 0 1 1 2 0 1 1 2 3 0 1 4 1 6 4 0 1 13 11 1 26 5 0 1 40 90 26 1 158 6 0 1 121 670 480 57 1 1330 7 0 1 364 4811 7870 2247 120 1 15414	Birch as a side of Liana and Ivy

illustrated examples:       Oak, Maple (seals with arity 5)       MapleMinor, Sycamore (blocks with adicity 4)

Maple(5, 0)=0 seals with arity 5 and depth 0 Oak(5, 0) = 1 with adicity 0 Maple(5, 0) = 0 with adicity 5 Maple(5, 1)=MapleMinor(4, 4)=1 ⋅ 6=16 seals with arity 5 and depth 1 Oak(5, 1) = 31 with adicities 1...5 Maple(5, 1) = 16 with adicity 5 seal blocks with arity 4 and depth 4 MapleMinor(4, 4) = Sycamore(4, 4) = 16 with adicity 4 Maple(5, 2)=MapleMinor(4, 3)=15 ⋅ 8=120 seals with arity 5 and depth 2 Oak(5, 2) = 155 with adicities 2...5 Maple(5, 2) = 120 with adicity 5 seal blocks with arity 4 and depth 3 MapleMinor(4, 3) = 120 with adicities 3...4 Sycamore(4, 3) = 112 with adicity 4 Maple(5, 3)=MapleMinor(4, 2)=35 ⋅ 4=140 seals with arity 5 and depth 3 Oak(5, 3) = 155 with adicities 3...5 Maple(5, 3) = 140 with adicity 5 seal blocks with arity 4 and depth 2 MapleMinor(4, 2) = 140 with adicities 2...4 Sycamore(4, 2) = 112 with adicity 4 Maple(5, 4)=MapleMinor(4, 1)=15 ⋅ 2=30 seals with arity 5 and depth 4 Oak(5, 4) = 31 with adicities 4...5 Maple(5, 4) = 30 with adicity 5 seal blocks with arity 4 and depth 1 MapleMinor(4, 1) = 30 with adicities 1...4 Sycamore(4, 1) = 16 with adicity 4 Maple(5, 5)=MapleMinor(4, 0)=1 ⋅ 1=1 seals with arity 5 and depth 5 Oak(5, 5) = Maple(5, 5) = 1 with adicity 5 seal blocks with arity 4 and depth 0 MapleMinor(4, 0) = 1 with adicity 0 Sycamore(4, 0) = 0 with adicity 4 triangles These are the same triangles as shown in the boxes above. They refer to the seals with arity 5. The main content of each image is a refinement of the shown slice of pyramid Ivy. Each column for a valency is refined into columns for each equivalence class. Shown above them are a slice of pyramid Liana and its refinement. Their entries are the column sums of the triangles below. depth 1 depth 2 depth 3 depth 4

number pyramids

The 🌊 wave pyramids Liana and Wisteria are refinements of triangle Oak.   In Liana the dimension valency is added. In Wisteria the dimension cohort weight.
Their second sides are the triangle Ash (for Liana) and its reflection (for Wisteria).

The 💧 drop pyramids Ivy and Lonicera are refinements of triangle Maple.   In Ivy the dimension valency is added. In Lonicera the dimension cohort weight.
Their second sides are the triangles Aspen (for Ivy) and Alder (for Lonicera).

seals a×d×v: pyramids Liana and Ivy

🌊    pyramid Liana

overview arity: top to bottom depth: back to front valency: left to right Indices in the image go from 1 to 7. Liana is always 1 where depth and valency are 0. But this column is not shown in the images. The sum ignoring valency is triangle Oak. The sum ignoring depth is triangle Ash. The layer sums (and row sums of these triangles) are sequence Daisy (A006116). fixed arity    (depth × valency matrices) The row sums are rows of triangle Oak. The column sums are rows of triangle Ash. The total sums are entries of Daisy. arity 0 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 Σ 1 1 arity 1 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 3 4 5 6 7 Σ 1 1 2 arity 2 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 3 2 1 1 3 4 5 6 7 Σ 1 2 2 5 arity 3 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 3 3 1 7 2 3 4 7 3 1 1 4 5 6 7 Σ 1 3 6 6 16 arity 4 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 4 6 4 1 15 2 6 16 13 35 3 4 11 15 4 1 1 5 6 7 Σ 1 4 12 24 26 67 arity 5 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 5 10 10 5 1 31 2 10 40 65 40 155 3 10 55 90 155 4 5 26 31 5 1 1 6 7 Σ 1 5 20 60 130 158 374 arity 6 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 6 15 20 15 6 1 63 2 15 80 195 240 121 651 3 20 165 540 670 1395 4 15 156 480 651 5 6 57 63 6 1 1 7 Σ 1 6 30 120 390 948 1330 2825 arity 7 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 7 21 35 35 21 7 1 127 2 21 140 455 840 847 364 2667 3 35 385 1890 4690 4811 11811 4 35 546 3360 7870 11811 5 21 399 2247 2667 6 7 120 127 7 1 1 Σ 1 7 42 210 910 3318 9310 15414 29212 fixed depth    (arity × valency matrices) The row sums are columns of triangle Oak. depth 0 v a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 1 depth 1 v a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 2 1 3 3 3 3 1 7 4 4 6 4 1 15 5 5 10 10 5 1 31 6 6 15 20 15 6 1 63 7 7 21 35 35 21 7 1 127 depth 2 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 3 4 7 4 6 16 13 35 5 10 40 65 40 155 6 15 80 195 240 121 651 7 21 140 455 840 847 364 2667 depth 3 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 4 11 15 5 10 55 90 155 6 20 165 540 670 1395 7 35 385 1890 4690 4811 11811 depth 4 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 5 26 31 6 15 156 480 651 7 35 546 3360 7870 11811 depth 5 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 6 57 63 7 21 399 2247 2667 depth 6 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 7 120 127 depth 7 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 sum:   triangle Ash v a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 2 1 2 2 5 3 1 3 6 6 16 4 1 4 12 24 26 67 5 1 5 20 60 130 158 374 6 1 6 30 120 390 948 1330 2825 7 1 7 42 210 910 3318 9310 15414 29212 fixed valency    (arity × depth matrices) The row sums are columns of triangle Ash. valency 0 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 1 valency 1 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 valency 2 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 2 3 3 3 6 4 6 6 12 5 10 10 20 6 15 15 30 7 21 21 42 valency 3 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 4 1 6 4 4 16 4 24 5 10 40 10 60 6 20 80 20 120 7 35 140 35 210 valency 4 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 13 11 1 26 5 5 65 55 5 130 6 15 195 165 15 390 7 35 455 385 35 910 valency 5 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 40 90 26 1 158 6 6 240 540 156 6 948 7 21 840 1890 546 21 3318 valency 6 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 121 670 480 57 1 1330 7 7 847 4690 3360 399 7 9310 valency 7 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 364 4811 7870 2247 120 1 15414 sum:   triangle Oak d a 0 1 2 3 4 5 6 7 Σ 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 other sides valency − depth = 0 Pascal's triangle   (with left column not shown) arity − valency = 0 triangle Birch

💧    pyramid Ivy

overview adicity: top to bottom depth: back to front valency: left to right Indices in the image go from 1 to 7. The entry Ivy(0, 0, 0) = 1 is not shown in the images. The sum ignoring valency is triangle Maple. The sum ignoring depth is triangle Aspen. The layer sums (and row sums of these triangles) are sequence Dahlia (A182176). The pyramid sides in the back (depth = 1) and front (valency − depth = 0) are Pascal's triangle. fixed adicity    (depth × valency matrices) The row sums are rows of triangle Maple. The column sums are rows of triangle Aspen. The total sums are entries of Dahlia. adicity 0 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 Σ 1 1 adicity 1 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 Σ 1 1 adicity 2 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 2 1 1 3 4 5 6 7 Σ 1 2 3 adicity 3 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 2 1 4 2 2 4 6 3 1 1 4 5 6 7 Σ 1 4 6 11 adicity 4 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 3 3 1 8 2 3 12 13 28 3 3 11 14 4 1 1 5 6 7 Σ 1 6 18 26 51 adicity 5 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 4 6 4 1 16 2 4 24 52 40 120 3 6 44 90 140 4 4 26 30 5 1 1 6 7 Σ 1 8 36 104 158 307 adicity 6 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 5 10 10 5 1 32 2 5 40 130 200 121 496 3 10 110 450 670 1240 4 10 130 480 620 5 5 57 62 6 1 1 7 Σ 1 10 60 260 790 1330 2451 adicity 7 v d 0 1 2 3 4 5 6 7 Σ 0 1 1 6 15 20 15 6 1 64 2 6 60 260 600 726 364 2016 3 15 220 1350 4020 4811 10416 4 20 390 2880 7870 11160 5 15 342 2247 2604 6 6 120 126 7 1 1 Σ 1 12 90 520 2370 7980 15414 26387 fixed depth    (adicity × valency matrices) The row sums are columns of triangle Maple. depth 0 v a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 depth 1 v a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 2 3 1 2 1 4 4 1 3 3 1 8 5 1 4 6 4 1 16 6 1 5 10 10 5 1 32 7 1 6 15 20 15 6 1 64 depth 2 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 2 4 6 4 3 12 13 28 5 4 24 52 40 120 6 5 40 130 200 121 496 7 6 60 260 600 726 364 2016 depth 3 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 3 11 14 5 6 44 90 140 6 10 110 450 670 1240 7 15 220 1350 4020 4811 10416 depth 4 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 4 26 30 6 10 130 480 620 7 20 390 2880 7870 11160 depth 5 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 5 57 62 7 15 342 2247 2604 depth 6 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 6 120 126 depth 7 v a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 sum:   triangle Aspen v a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 1 2 3 3 1 4 6 11 4 1 6 18 26 51 5 1 8 36 104 158 307 6 1 10 60 260 790 1330 2451 7 1 12 90 520 2370 7980 15414 26387 fixed valency    (adicity × depth matrices) The row sums are columns of triangle Aspen. valency 0 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 valency 1 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 1 valency 2 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 2 3 2 2 4 4 3 3 6 5 4 4 8 6 5 5 10 7 6 6 12 valency 3 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 4 1 6 4 3 12 3 18 5 6 24 6 36 6 10 40 10 60 7 15 60 15 90 valency 4 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 13 11 1 26 5 4 52 44 4 104 6 10 130 110 10 260 7 20 260 220 20 520 valency 5 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 40 90 26 1 158 6 5 200 450 130 5 790 7 15 600 1350 390 15 2370 valency 6 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 121 670 480 57 1 1330 7 6 726 4020 2880 342 6 7980 valency 7 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 364 4811 7870 2247 120 1 15414 sum:   triangle Maple d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 2 1 3 3 4 6 1 11 4 8 28 14 1 51 5 16 120 140 30 1 307 6 32 496 1240 620 62 1 2451 7 64 2016 10416 11160 2604 126 1 26387 other sides valency − depth = 0 Pascal's triangle   (trivial column on the left not shown) adicity − valency = 0 triangle Birch

seals a×d×c: pyramids Wisteria and Lonicera

🌊    pyramid Wisteria

overview arity: top to bottom depth: back to front cohort weight: left to right The sum ignoring cohort weight is triangle Oak. The sum ignoring depth is reflected Ash. The layer sums (and row sums of these triangles) are sequence Daisy (A006116). fixed arity    (depth × cohort weight matrices) The row sums are rows of Oak. The column sums are rows of reflected Ash. The total sums are entries of Daisy. arity 0 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 Σ 1 1 arity 1 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 3 4 5 6 7 Σ 1 1 2 arity 2 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 2 3 2 1 1 3 4 5 6 7 Σ 2 2 1 5 arity 3 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 4 3 7 2 1 3 3 7 3 1 1 4 5 6 7 Σ 6 6 3 1 16 arity 4 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 11 4 15 2 13 16 6 35 3 1 4 6 4 15 4 1 1 5 6 7 Σ 26 24 12 4 1 67 arity 5 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 26 5 31 2 90 55 10 155 3 40 65 40 10 155 4 1 5 10 10 5 31 5 1 1 6 7 Σ 158 130 60 20 5 1 374 arity 6 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 57 6 63 2 480 156 15 651 3 670 540 165 20 1395 4 121 240 195 80 15 651 5 1 6 15 20 15 6 63 6 1 1 7 Σ 1330 948 390 120 30 6 1 2825 arity 7 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 120 7 127 2 2247 399 21 2667 3 7870 3360 546 35 11811 4 4811 4690 1890 385 35 11811 5 364 847 840 455 140 21 2667 6 1 7 21 35 35 21 7 127 7 1 1 Σ 15414 9310 3318 910 210 42 7 1 29212 fixed depth    (arity × cohort weight matrices) The row sums are columns of Oak. depth 0 c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 1 depth 1 The left column shows the Euler numbers (A000295). Their positive terms are summations of 1, 3, 7, 15, 31... c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 2 3 3 4 3 7 4 11 4 15 5 26 5 31 6 57 6 63 7 120 7 127 depth 2 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 1 3 3 7 4 13 16 6 35 5 90 55 10 155 6 480 156 15 651 7 2247 399 21 2667 depth 3 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 1 4 6 4 15 5 40 65 40 10 155 6 670 540 165 20 1395 7 7870 3360 546 35 11811 depth 4 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 1 5 10 10 5 31 6 121 240 195 80 15 651 7 4811 4690 1890 385 35 11811 depth 5 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 1 6 15 20 15 6 63 7 364 847 840 455 140 21 2667 depth 6 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 1 7 21 35 35 21 7 127 depth 7 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 sum:   reflected triangle Ash c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 2 2 2 1 5 3 6 6 3 1 16 4 26 24 12 4 1 67 5 158 130 60 20 5 1 374 6 1330 948 390 120 30 6 1 2825 7 15414 9310 3318 910 210 42 7 1 29212 fixed cohort weight    (arity × depth matrices) The row sums are columns of reflected Ash. cohort weight 0 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 1 1 2 3 1 4 1 6 4 1 11 13 1 26 5 1 26 90 40 1 158 6 1 57 480 670 121 1 1330 7 1 120 2247 7870 4811 364 1 15414 cohort weight 1 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 2 2 3 3 3 6 4 4 16 4 24 5 5 55 65 5 130 6 6 156 540 240 6 948 7 7 399 3360 4690 847 7 9310 cohort weight 2 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 3 3 4 6 6 12 5 10 40 10 60 6 15 165 195 15 390 7 21 546 1890 840 21 3318 cohort weight 3 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 4 4 5 10 10 20 6 20 80 20 120 7 35 385 455 35 910 cohort weight 4 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 5 5 6 15 15 30 7 35 140 35 210 cohort weight 5 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 6 6 7 21 21 42 cohort weight 6 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 7 7 cohort weight 7 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 sum:   triangle Oak d a 0 1 2 3 4 5 6 7 Σ 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 other sides depth − cohort weight = 0 Pascal's triangle adicity − depth = 1 almost Pascal's triangle   (without right diagonal)

💧    pyramid Lonicera

overview adicity: top to bottom depth: back to front cohort weight: left to right The sum ignoring cohort weight is triangle Maple. The sum ignoring depth is triangle Alder. The layer sums (and row sums of these triangles) are sequence Dahlia (A182176). fixed adicity    (depth × cohort weight matrices) The row sums are rows of Maple. The column sums are rows of Alder. The total sums are entries of Dahlia. adicity 0 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 Σ 1 1 adicity 1 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 Σ 1 1 adicity 2 c d 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 2 1 1 3 4 5 6 7 Σ 1 1 1 3 adicity 3 c d 0 1 2 3 4 5 6 7 Σ 0 1 3 1 4 2 1 3 2 6 3 1 1 4 5 6 7 Σ 4 4 2 1 11 adicity 4 c d 0 1 2 3 4 5 6 7 Σ 0 1 7 1 8 2 12 13 3 28 3 1 4 6 3 14 4 1 1 5 6 7 Σ 20 18 9 3 1 51 adicity 5 c d 0 1 2 3 4 5 6 7 Σ 0 1 15 1 16 2 77 39 4 120 3 39 61 34 6 140 4 1 5 10 10 4 30 5 1 1 6 7 Σ 132 106 48 16 4 1 307 adicity 6 c d 0 1 2 3 4 5 6 7 Σ 0 1 31 1 32 2 390 101 5 496 3 630 475 125 10 1240 4 120 235 185 70 10 620 5 1 6 15 20 15 5 62 6 1 1 7 Σ 1172 818 330 100 25 5 1 2451 adicity 7 c d 0 1 2 3 4 5 6 7 Σ 0 1 63 1 64 2 1767 243 6 2016 3 7200 2820 381 15 10416 4 4690 4450 1695 305 20 11160 5 363 841 825 435 125 15 2604 6 1 7 21 35 35 21 6 126 7 1 1 Σ 14084 8362 2928 790 180 36 6 1 26387 fixed depth    (adicity × cohort weight matrices) The row sums are columns of Maple. depth 0 c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 3 4 5 6 7 depth 1 c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 2 3 3 1 4 4 7 1 8 5 15 1 16 6 31 1 32 7 63 1 64 depth 2 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 1 3 2 6 4 12 13 3 28 5 77 39 4 120 6 390 101 5 496 7 1767 243 6 2016 depth 3 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 1 4 6 3 14 5 39 61 34 6 140 6 630 475 125 10 1240 7 7200 2820 381 15 10416 depth 4 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 1 5 10 10 4 30 6 120 235 185 70 10 620 7 4690 4450 1695 305 20 11160 depth 5 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 1 6 15 20 15 5 62 7 363 841 825 435 125 15 2604 depth 6 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 1 7 21 35 35 21 6 126 depth 7 c a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 sum:   triangle Alder c a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 1 1 1 3 3 4 4 2 1 11 4 20 18 9 3 1 51 5 132 106 48 16 4 1 307 6 1172 818 330 100 25 5 1 2451 7 14084 8362 2928 790 180 36 6 1 26387 fixed cohort weight    (adicity × depth matrices) The row sums are columns of Alder. cohort weight 0 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 3 3 1 4 4 7 12 1 20 5 15 77 39 1 132 6 31 390 630 120 1 1172 7 63 1767 7200 4690 363 1 14084 cohort weight 1 d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 2 1 1 3 1 3 4 4 1 13 4 18 5 1 39 61 5 106 6 1 101 475 235 6 818 7 1 243 2820 4450 841 7 8362 cohort weight 2 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 1 1 3 2 2 4 3 6 9 5 4 34 10 48 6 5 125 185 15 330 7 6 381 1695 825 21 2928 cohort weight 3 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 1 1 4 3 3 5 6 10 16 6 10 70 20 100 7 15 305 435 35 790 cohort weight 4 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 1 1 5 4 4 6 10 15 25 7 20 125 35 180 cohort weight 5 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 1 1 6 5 5 7 15 21 36 cohort weight 6 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 1 1 7 6 6 cohort weight 7 d a 0 1 2 3 4 5 6 7 Σ 0 1 2 3 4 5 6 7 1 1 File:Number pyramid Lonicera; cohort weight 7.png sum:   triangle Maple d a 0 1 2 3 4 5 6 7 Σ 0 1 1 1 1 1 2 2 1 3 3 4 6 1 11 4 8 28 14 1 51 5 16 120 140 30 1 307 6 32 496 1240 620 62 1 2451 7 64 2016 10416 11160 2604 126 1 26387 other sides depth − cohort weight = 0 Pascal's triangle   (with trivial column on the left) adicity − depth = 1 almost Pascal's triangle   (without right diagonal, next reduced by 1)