# Pyramid Liana

These pyramid pairs are refinements of the triangle pair /$_{Maple}$. In $^{Liana}$/$_{Ivy}$ the axis valency is added. In $^{TwistedLiana}$/$_{Lonicera}$ the axis gravity is added.
🌊 Liana and TwistedLiana are refinements of Oak.  Their sums along axis depth are Ash and TwistedAsh.
💧 Ivy and Lonicera are refinements of Maple.  Their sums along axis depth are Aspen and Alder.
Liana and TwistedLiana are essentially the same pyramid: ${\text{Liana}}(a,x,y)={\text{TwistedLiana}}(a,a-x,a-y)$ So ${\text{Liana}}(a,x,y)$ is the cardinality of two sets of $a$-ary seals: Those with depth $x$ and valency $y$, and those with depth $a-x$ and gravity $a-y$.

## Pyramids Liana and Ivy

Liana(a, d, v) is the number of seals with arity a, depth d and valency v. Ivy(a, d, v) is the number of seals with adicity a, depth d and valency v.
🌊    pyramid Liana :
fixed arity    (depth × valency matrices) :
arity 0 :
| vd | 0 | Σ |
| 0 | 1 | 1 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
| Σ | 1 | 1 |
arity 1 :
| vd | 0 | 1 | Σ |
| 0 | 1 | | 1 |
| 1 | | 1 | 1 |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | | | |
| 7 | | | |
| Σ | 1 | 1 | 2 |
arity 2 :
| vd | 0 | 1 | 2 | Σ |
| 0 | 1 | | | 1 |
| 1 | | 2 | 1 | 3 |
| 2 | | | 1 | 1 |
| 3 | | | | |
| 4 | | | | |
| 5 | | | | |
| 6 | | | | |
| 7 | | | | |
| Σ | 1 | 2 | 2 | 5 |
arity 3 :
| vd | 0 | 1 | 2 | 3 | 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 :
| vd | 0 | 1 | 2 | 3 | 4 | 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 :
| vd | 0 | 1 | 2 | 3 | 4 | 5 | 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 :
| vd | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 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 :
| vd | 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) :
depth 0 :
| va | 0 | 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 :
| va | 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 :
| va | 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 :
| va | 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 :
| va | 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 :
| va | 5 | 6 | 7 | Σ |
| 0 | | | | |
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | | | | |
| 5 | 1 | | | 1 |
| 6 | 6 | 57 | | 63 |
| 7 | 21 | 399 | 2247 | 2667 |
depth 6 :
| va | 6 | 7 | Σ |
| 0 | | | |
| 1 | | | |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | 1 | | 1 |
| 7 | 7 | 120 | 127 |
depth 7 :
| va | 7 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | 1 | 1 |
sum:   triangle Ash :
| va | 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) :
valency 0 :
| da | 0 | 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 :
| da | 1 | 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 :
| da | 1 | 2 | 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 :
| da | 1 | 2 | 3 | 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 :
| da | 1 | 2 | 3 | 4 | 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 :
| da | 1 | 2 | 3 | 4 | 5 | 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 :
| da | 1 | 2 | 3 | 4 | 5 | 6 | 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 :
| da | 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 :
| da | 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 |
💧    pyramid Ivy :
fixed adicity    (depth × valency matrices) :
adicity 0 :
| vd | 0 | Σ |
| 0 | 1 | 1 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
| Σ | 1 | 1 |
adicity 1 :
| vd | 1 | Σ |
| 0 | | |
| 1 | 1 | 1 |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
| Σ | 1 | 1 |
adicity 2 :
| vd | 1 | 2 | Σ |
| 0 | | | |
| 1 | 1 | 1 | 2 |
| 2 | | 1 | 1 |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | | | |
| 7 | | | |
| Σ | 1 | 2 | 3 |
adicity 3 :
| vd | 1 | 2 | 3 | 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 :
| vd | 1 | 2 | 3 | 4 | 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 :
| vd | 1 | 2 | 3 | 4 | 5 | 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 :
| vd | 1 | 2 | 3 | 4 | 5 | 6 | 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 :
| vd | 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) :
depth 0 :
| va | 0 | Σ |
| 0 | 1 | 1 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
depth 1 :
| va | 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 :
| va | 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 :
| va | 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 :
| va | 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 :
| va | 5 | 6 | 7 | Σ |
| 0 | | | | |
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | | | | |
| 5 | 1 | | | 1 |
| 6 | 5 | 57 | | 62 |
| 7 | 15 | 342 | 2247 | 2604 |
depth 6 :
| va | 6 | 7 | Σ |
| 0 | | | |
| 1 | | | |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | 1 | | 1 |
| 7 | 6 | 120 | 126 |
depth 7 :
| va | 7 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | 1 | 1 |
sum:   triangle Aspen :
| va | 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) :
valency 0 :
| da | 0 | Σ |
| 0 | 1 | 1 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
valency 1 :
| da | 1 | 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 :
| da | 1 | 2 | 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 :
| da | 1 | 2 | 3 | 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 :
| da | 1 | 2 | 3 | 4 | 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 :
| da | 1 | 2 | 3 | 4 | 5 | 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 :
| da | 1 | 2 | 3 | 4 | 5 | 6 | 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 :
| da | 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 :
| da | 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 |

## Pyramids TwistedLiana and Lonicera

TwistedLiana(a, d, g) is the number of seals with arity a, depth d and gravity g. Lonicera(a, d, g) is the number of seals with adicity a, depth d and gravity g.
🌊    pyramid TwistedLiana :
fixed arity    (depth × gravity matrices) :
arity 0 :
| gd | 0 | Σ |
| 0 | 1 | 1 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
| Σ | 1 | 1 |
arity 1 :
| gd | 0 | 1 | Σ |
| 0 | 1 | | 1 |
| 1 | | 1 | 1 |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | | | |
| 7 | | | |
| Σ | 1 | 1 | 2 |
arity 2 :
| gd | 0 | 1 | 2 | Σ |
| 0 | 1 | | | 1 |
| 1 | 1 | 2 | | 3 |
| 2 | | | 1 | 1 |
| 3 | | | | |
| 4 | | | | |
| 5 | | | | |
| 6 | | | | |
| 7 | | | | |
| Σ | 2 | 2 | 1 | 5 |
arity 3 :
| gd | 0 | 1 | 2 | 3 | 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 :
| gd | 0 | 1 | 2 | 3 | 4 | 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 :
| gd | 0 | 1 | 2 | 3 | 4 | 5 | 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 :
| gd | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 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 :
| gd | 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 × gravity matrices) :
depth 0 :
| ga | 0 | 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... :
| ga | 0 | 1 | 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 :
| ga | 0 | 1 | 2 | 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 :
| ga | 0 | 1 | 2 | 3 | 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 :
| ga | 0 | 1 | 2 | 3 | 4 | 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 :
| ga | 0 | 1 | 2 | 3 | 4 | 5 | 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 :
| ga | 0 | 1 | 2 | 3 | 4 | 5 | 6 | Σ |
| 0 | | | | | | | | |
| 1 | | | | | | | | |
| 2 | | | | | | | | |
| 3 | | | | | | | | |
| 4 | | | | | | | | |
| 5 | | | | | | | | |
| 6 | | | | | | | 1 | 1 |
| 7 | 1 | 7 | 21 | 35 | 35 | 21 | 7 | 127 |
depth 7 :
| ga | 7 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | 1 | 1 |
sum:   triangle TwistedAsh :
| ga | 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 gravity    (arity × depth matrices) :
gravity 0 :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 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 |
gravity 1 :
| da | 1 | 2 | 3 | 4 | 5 | 6 | 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 |
gravity 2 :
| da | 2 | 3 | 4 | 5 | 6 | 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 |
gravity 3 :
| da | 3 | 4 | 5 | 6 | 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 |
gravity 4 :
| da | 4 | 5 | 6 | Σ |
| 0 | | | | |
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | 1 | | | 1 |
| 5 | 5 | | | 5 |
| 6 | 15 | 15 | | 30 |
| 7 | 35 | 140 | 35 | 210 |
gravity 5 :
| da | 5 | 6 | Σ |
| 0 | | | |
| 1 | | | |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | 1 | | 1 |
| 6 | 6 | | 6 |
| 7 | 21 | 21 | 42 |
gravity 6 :
| da | 6 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | 1 | 1 |
| 7 | 7 | 7 |
gravity 7 :
| da | 7 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | 1 | 1 |
sum:   triangle Oak :
| da | 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 |
💧    pyramid Lonicera :
fixed adicity    (depth × gravity matrices) :
adicity 0 :
| gd | 0 | Σ |
| 0 | 1 | 1 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
| Σ | 1 | 1 |
adicity 1 :
| gd | 1 | Σ |
| 0 | | |
| 1 | 1 | 1 |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
| Σ | 1 | 1 |
adicity 2 :
| gd | 0 | 1 | 2 | Σ |
| 0 | | | | |
| 1 | 1 | 1 | | 2 |
| 2 | | | 1 | 1 |
| 3 | | | | |
| 4 | | | | |
| 5 | | | | |
| 6 | | | | |
| 7 | | | | |
| Σ | 1 | 1 | 1 | 3 |
adicity 3 :
| gd | 0 | 1 | 2 | 3 | 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 :
| gd | 0 | 1 | 2 | 3 | 4 | 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 :
| gd | 0 | 1 | 2 | 3 | 4 | 5 | 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 :
| gd | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 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 :
| gd | 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 × gravity matrices) :
depth 0 :
| ga | 0 | Σ |
| 0 | 1 | 1 |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | | |
depth 1 :
| ga | 0 | 1 | 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 :
| ga | 0 | 1 | 2 | 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 :
| ga | 0 | 1 | 2 | 3 | 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 :
| ga | 0 | 1 | 2 | 3 | 4 | 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 :
| ga | 0 | 1 | 2 | 3 | 4 | 5 | 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 :
| ga | 0 | 1 | 2 | 3 | 4 | 5 | 6 | Σ |
| 0 | | | | | | | | |
| 1 | | | | | | | | |
| 2 | | | | | | | | |
| 3 | | | | | | | | |
| 4 | | | | | | | | |
| 5 | | | | | | | | |
| 6 | | | | | | | 1 | 1 |
| 7 | 1 | 7 | 21 | 35 | 35 | 21 | 6 | 126 |
depth 7 :
| ga | 7 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | 1 | 1 |
sum:   triangle Alder :
| ga | 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 gravity    (adicity × depth matrices) :
gravity 0 :
| da | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 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 |
gravity 1 :
| da | 1 | 2 | 3 | 4 | 5 | 6 | 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 |
gravity 2 :
| da | 2 | 3 | 4 | 5 | 6 | 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 |
gravity 3 :
| da | 3 | 4 | 5 | 6 | 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 |
gravity 4 :
| da | 4 | 5 | 6 | Σ |
| 0 | | | | |
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | 1 | | | 1 |
| 5 | 4 | | | 4 |
| 6 | 10 | 15 | | 25 |
| 7 | 20 | 125 | 35 | 180 |
gravity 5 :
| da | 5 | 6 | Σ |
| 0 | | | |
| 1 | | | |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | 1 | | 1 |
| 6 | 5 | | 5 |
| 7 | 15 | 21 | 36 |
gravity 6 :
| da | 6 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | 1 | 1 |
| 7 | 6 | 6 |
gravity 7 :
| da | 7 | Σ |
| 0 | | |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
| 6 | | |
| 7 | 1 | 1 |
sum:   triangle Maple :
| da | 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 |