Differentials of the rapacious rule
The rule as we understood it
$d_r\colon E_r(n,s,c)\to E_r(n-1,\,s+r,\,c+1)$; on each page $r=2,3,\dots$ the maximal rank the dimensions allow is taken. The pages are processed in order.
The grammar gives $\text{stem}=4\sum(\text{indices})-\#\lambda$, that is, $c\equiv \text{stem}+s \pmod 4$. Hence an arrow “colour $+1$” is possible only for $r\equiv2\pmod4$: $d_2, d_6, d_{10},\dots$ The variant “colour $-1$” gives no differentials at all, and in it $\pi_{44}$ keeps four classes, whereas the lecture says it is empty.
Where the rule is ambiguous
On page $r$ the possible arrows line up into chains $x\to y\to z\to\cdots$ with decreasing $n$. The maximal total rank on a chain is determined uniquely, but which classes die is not: a middle class can be either a target or a source. “low” fills a chain starting from the end with the smaller $n$, “high” from the end with the larger. For $n\le149$ the two variants differ only here:
only in “low”:
| from (s, c) | to (s, c) | rank | |
|---|---|---|---|
| $d_{2}$ | $\pi_{42}$ (7, 2) | $\pi_{41}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{52}$ (9, 2) | $\pi_{51}$ (11, 3) | 1 |
| $d_{2}$ | $\pi_{62}$ (11, 2) | $\pi_{61}$ (13, 3) | 1 |
| $d_{2}$ | $\pi_{72}$ (13, 2) | $\pi_{71}$ (15, 3) | 1 |
| $d_{2}$ | $\pi_{75}$ (6, 2) | $\pi_{74}$ (8, 3) | 1 |
| $d_{2}$ | $\pi_{78}$ (7, 2) | $\pi_{77}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{79}$ (7, 3) | $\pi_{78}$ (9, 4) | 1 |
| $d_{2}$ | $\pi_{82}$ (15, 2) | $\pi_{81}$ (17, 3) | 1 |
| $d_{2}$ | $\pi_{85}$ (8, 2) | $\pi_{84}$ (10, 3) | 1 |
| $d_{2}$ | $\pi_{86}$ (7, 2) | $\pi_{85}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{88}$ (9, 2) | $\pi_{87}$ (11, 3) | 1 |
| $d_{2}$ | $\pi_{89}$ (9, 3) | $\pi_{88}$ (11, 4) | 1 |
| $d_{2}$ | $\pi_{90}$ (10, 5) | $\pi_{89}$ (12, 6) | 1 |
| $d_{2}$ | $\pi_{92}$ (17, 2) | $\pi_{91}$ (19, 3) | 1 |
| $d_{2}$ | $\pi_{93}$ (8, 2) | $\pi_{92}$ (10, 3) | 1 |
| $d_{2}$ | $\pi_{94}$ (8, 3) | $\pi_{93}$ (10, 4) | 1 |
| $d_{2}$ | $\pi_{95}$ (10, 2) | $\pi_{94}$ (12, 3) | 1 |
| $d_{2}$ | $\pi_{97}$ (9, 3) | $\pi_{96}$ (11, 4) | 1 |
| $d_{2}$ | $\pi_{98}$ (9, 4) | $\pi_{97}$ (11, 5) | 1 |
| $d_{2}$ | $\pi_{98}$ (11, 2) | $\pi_{97}$ (13, 3) | 1 |
| $d_{2}$ | $\pi_{99}$ (10, 6) | $\pi_{98}$ (12, 7) | 1 |
| $d_{2}$ | $\pi_{99}$ (11, 3) | $\pi_{98}$ (13, 4) | 1 |
| $d_{2}$ | $\pi_{100}$ (12, 5) | $\pi_{99}$ (14, 6) | 1 |
| $d_{2}$ | $\pi_{102}$ (8, 3) | $\pi_{101}$ (10, 4) | 1 |
| $d_{2}$ | $\pi_{102}$ (19, 2) | $\pi_{101}$ (21, 3) | 1 |
| $d_{2}$ | $\pi_{109}$ (8, 2) | $\pi_{108}$ (10, 3) | 1 |
| $d_{2}$ | $\pi_{108}$ (11, 4) | $\pi_{107}$ (13, 5) | 1 |
| $d_{2}$ | $\pi_{110}$ (7, 2) | $\pi_{109}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{110}$ (14, 5) | $\pi_{109}$ (16, 6) | 1 |
| $d_{2}$ | $\pi_{112}$ (21, 2) | $\pi_{111}$ (23, 3) | 1 |
| $d_{2}$ | $\pi_{114}$ (12, 3) | $\pi_{113}$ (14, 4) | 1 |
| $d_{2}$ | $\pi_{116}$ (8, 1) | $\pi_{115}$ (10, 2) | 1 |
| $d_{2}$ | $\pi_{118}$ (13, 4) | $\pi_{117}$ (15, 5) | 1 |
| $d_{2}$ | $\pi_{120}$ (9, 2) | $\pi_{119}$ (11, 3) | 1 |
| $d_{2}$ | $\pi_{120}$ (16, 5) | $\pi_{119}$ (18, 6) | 1 |
| $d_{2}$ | $\pi_{122}$ (23, 2) | $\pi_{121}$ (25, 3) | 1 |
| $d_{2}$ | $\pi_{128}$ (15, 4) | $\pi_{127}$ (17, 5) | 1 |
| $d_{2}$ | $\pi_{130}$ (18, 5) | $\pi_{129}$ (20, 6) | 1 |
| $d_{2}$ | $\pi_{132}$ (25, 2) | $\pi_{131}$ (27, 3) | 1 |
| $d_{2}$ | $\pi_{133}$ (11, 5) | $\pi_{132}$ (13, 6) | 1 |
| $d_{2}$ | $\pi_{134}$ (10, 5) | $\pi_{133}$ (12, 6) | 1 |
| $d_{2}$ | $\pi_{137}$ (12, 6) | $\pi_{136}$ (14, 7) | 1 |
| $d_{2}$ | $\pi_{138}$ (13, 8) | $\pi_{137}$ (15, 9) | 1 |
| $d_{2}$ | $\pi_{138}$ (17, 4) | $\pi_{137}$ (19, 5) | 1 |
| $d_{2}$ | $\pi_{140}$ (20, 5) | $\pi_{139}$ (22, 6) | 1 |
| $d_{2}$ | $\pi_{141}$ (11, 5) | $\pi_{140}$ (13, 6) | 1 |
| $d_{2}$ | $\pi_{142}$ (11, 6) | $\pi_{141}$ (13, 7) | 1 |
| $d_{2}$ | $\pi_{142}$ (27, 2) | $\pi_{141}$ (29, 3) | 1 |
| $d_{2}$ | $\pi_{143}$ (13, 5) | $\pi_{142}$ (15, 6) | 1 |
| $d_{2}$ | $\pi_{145}$ (9, 3) | $\pi_{144}$ (11, 4) | 1 |
| $d_{2}$ | $\pi_{145}$ (12, 6) | $\pi_{144}$ (14, 7) | 1 |
| $d_{2}$ | $\pi_{146}$ (12, 7) | $\pi_{145}$ (14, 8) | 1 |
| $d_{2}$ | $\pi_{146}$ (14, 5) | $\pi_{145}$ (16, 6) | 1 |
| $d_{2}$ | $\pi_{148}$ (10, 3) | $\pi_{147}$ (12, 4) | 1 |
| $d_{2}$ | $\pi_{148}$ (19, 4) | $\pi_{147}$ (21, 5) | 1 |
| $d_{2}$ | $\pi_{147}$ (14, 6) | $\pi_{146}$ (16, 7) | 1 |
| $d_{2}$ | $\pi_{149}$ (10, 4) | $\pi_{148}$ (12, 5) | 1 |
| $d_{2}$ | $\pi_{149}$ (21, 3) | $\pi_{148}$ (23, 4) | 1 |
| $d_{6}$ | $\pi_{84}$ (7, 0) | $\pi_{83}$ (13, 1) | 1 |
| $d_{6}$ | $\pi_{94}$ (9, 0) | $\pi_{93}$ (15, 1) | 1 |
| $d_{6}$ | $\pi_{104}$ (11, 0) | $\pi_{103}$ (17, 1) | 1 |
| $d_{6}$ | $\pi_{114}$ (13, 0) | $\pi_{113}$ (19, 1) | 1 |
| $d_{6}$ | $\pi_{122}$ (8, 3) | $\pi_{121}$ (14, 4) | 1 |
| $d_{6}$ | $\pi_{124}$ (15, 0) | $\pi_{123}$ (21, 1) | 1 |
| $d_{6}$ | $\pi_{132}$ (10, 3) | $\pi_{131}$ (16, 4) | 1 |
| $d_{6}$ | $\pi_{134}$ (17, 0) | $\pi_{133}$ (23, 1) | 1 |
| $d_{6}$ | $\pi_{142}$ (12, 3) | $\pi_{141}$ (18, 4) | 1 |
| $d_{6}$ | $\pi_{144}$ (19, 0) | $\pi_{143}$ (25, 1) | 1 |
| $d_{14}$ | $\pi_{117}$ (6, 0) | $\pi_{116}$ (20, 1) | 1 |
only in “high”:
| from (s, c) | to (s, c) | rank | |
|---|---|---|---|
| $d_{2}$ | $\pi_{43}$ (5, 1) | $\pi_{42}$ (7, 2) | 1 |
| $d_{2}$ | $\pi_{53}$ (7, 1) | $\pi_{52}$ (9, 2) | 1 |
| $d_{2}$ | $\pi_{63}$ (9, 1) | $\pi_{62}$ (11, 2) | 1 |
| $d_{2}$ | $\pi_{73}$ (11, 1) | $\pi_{72}$ (13, 2) | 1 |
| $d_{2}$ | $\pi_{76}$ (4, 1) | $\pi_{75}$ (6, 2) | 1 |
| $d_{2}$ | $\pi_{79}$ (5, 1) | $\pi_{78}$ (7, 2) | 1 |
| $d_{2}$ | $\pi_{80}$ (5, 2) | $\pi_{79}$ (7, 3) | 1 |
| $d_{2}$ | $\pi_{83}$ (13, 1) | $\pi_{82}$ (15, 2) | 1 |
| $d_{2}$ | $\pi_{86}$ (6, 1) | $\pi_{85}$ (8, 2) | 1 |
| $d_{2}$ | $\pi_{87}$ (5, 1) | $\pi_{86}$ (7, 2) | 1 |
| $d_{2}$ | $\pi_{89}$ (7, 1) | $\pi_{88}$ (9, 2) | 1 |
| $d_{2}$ | $\pi_{90}$ (7, 2) | $\pi_{89}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{93}$ (15, 1) | $\pi_{92}$ (17, 2) | 1 |
| $d_{2}$ | $\pi_{94}$ (6, 1) | $\pi_{93}$ (8, 2) | 1 |
| $d_{2}$ | $\pi_{95}$ (6, 2) | $\pi_{94}$ (8, 3) | 1 |
| $d_{2}$ | $\pi_{96}$ (8, 1) | $\pi_{95}$ (10, 2) | 1 |
| $d_{2}$ | $\pi_{98}$ (7, 2) | $\pi_{97}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{99}$ (7, 3) | $\pi_{98}$ (9, 4) | 1 |
| $d_{2}$ | $\pi_{99}$ (9, 1) | $\pi_{98}$ (11, 2) | 1 |
| $d_{2}$ | $\pi_{100}$ (8, 5) | $\pi_{99}$ (10, 6) | 1 |
| $d_{2}$ | $\pi_{100}$ (9, 2) | $\pi_{99}$ (11, 3) | 1 |
| $d_{2}$ | $\pi_{103}$ (6, 2) | $\pi_{102}$ (8, 3) | 1 |
| $d_{2}$ | $\pi_{103}$ (17, 1) | $\pi_{102}$ (19, 2) | 1 |
| $d_{2}$ | $\pi_{110}$ (6, 1) | $\pi_{109}$ (8, 2) | 1 |
| $d_{2}$ | $\pi_{111}$ (5, 1) | $\pi_{110}$ (7, 2) | 1 |
| $d_{2}$ | $\pi_{109}$ (9, 3) | $\pi_{108}$ (11, 4) | 1 |
| $d_{2}$ | $\pi_{113}$ (19, 1) | $\pi_{112}$ (21, 2) | 1 |
| $d_{2}$ | $\pi_{117}$ (6, 0) | $\pi_{116}$ (8, 1) | 1 |
| $d_{2}$ | $\pi_{115}$ (10, 2) | $\pi_{114}$ (12, 3) | 1 |
| $d_{2}$ | $\pi_{121}$ (7, 1) | $\pi_{120}$ (9, 2) | 1 |
| $d_{2}$ | $\pi_{119}$ (11, 3) | $\pi_{118}$ (13, 4) | 1 |
| $d_{2}$ | $\pi_{123}$ (21, 1) | $\pi_{122}$ (23, 2) | 1 |
| $d_{2}$ | $\pi_{129}$ (13, 3) | $\pi_{128}$ (15, 4) | 1 |
| $d_{2}$ | $\pi_{133}$ (23, 1) | $\pi_{132}$ (25, 2) | 1 |
| $d_{2}$ | $\pi_{134}$ (9, 4) | $\pi_{133}$ (11, 5) | 1 |
| $d_{2}$ | $\pi_{135}$ (8, 4) | $\pi_{134}$ (10, 5) | 1 |
| $d_{2}$ | $\pi_{138}$ (10, 5) | $\pi_{137}$ (12, 6) | 1 |
| $d_{2}$ | $\pi_{139}$ (11, 7) | $\pi_{138}$ (13, 8) | 1 |
| $d_{2}$ | $\pi_{139}$ (15, 3) | $\pi_{138}$ (17, 4) | 1 |
| $d_{2}$ | $\pi_{142}$ (9, 4) | $\pi_{141}$ (11, 5) | 1 |
| $d_{2}$ | $\pi_{143}$ (9, 5) | $\pi_{142}$ (11, 6) | 1 |
| $d_{2}$ | $\pi_{143}$ (25, 1) | $\pi_{142}$ (27, 2) | 1 |
| $d_{2}$ | $\pi_{146}$ (7, 2) | $\pi_{145}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{144}$ (11, 4) | $\pi_{143}$ (13, 5) | 1 |
| $d_{2}$ | $\pi_{146}$ (10, 5) | $\pi_{145}$ (12, 6) | 1 |
| $d_{2}$ | $\pi_{147}$ (10, 6) | $\pi_{146}$ (12, 7) | 1 |
| $d_{2}$ | $\pi_{149}$ (8, 2) | $\pi_{148}$ (10, 3) | 1 |
| $d_{2}$ | $\pi_{147}$ (12, 4) | $\pi_{146}$ (14, 5) | 1 |
| $d_{2}$ | $\pi_{149}$ (17, 3) | $\pi_{148}$ (19, 4) | 1 |
| $d_{2}$ | $\pi_{150}$ (8, 3) | $\pi_{149}$ (10, 4) | 1 |
| $d_{2}$ | $\pi_{148}$ (12, 5) | $\pi_{147}$ (14, 6) | 1 |
| $d_{2}$ | $\pi_{150}$ (19, 2) | $\pi_{149}$ (21, 3) | 1 |
| $d_{6}$ | $\pi_{52}$ (5, 2) | $\pi_{51}$ (11, 3) | 1 |
| $d_{10}$ | $\pi_{148}$ (11, 4) | $\pi_{147}$ (21, 5) | 1 |
| $d_{10}$ | $\pi_{149}$ (13, 3) | $\pi_{148}$ (23, 4) | 1 |
Where the variants differ and Toda's value is known ($n\le82$): $\pi_{41}$: low 1, high 2, Toda 1 — as low; $\pi_{43}$: low 1, high 0, Toda 1 — as low; $\pi_{52}$: low 2, high 1, Toda 2 — as low; $\pi_{53}$: low 3, high 2, Toda 3 — as low; $\pi_{61}$: low 2, high 3, Toda 2 — as low; $\pi_{63}$: low 3, high 2, Toda 2 — as high; $\pi_{71}$: low 1, high 2, Toda 1 — as low; $\pi_{73}$: low 3, high 2, Toda 1 — neither; $\pi_{74}$: low 3, high 4, Toda 1 — neither; $\pi_{76}$: low 4, high 3, Toda 0 — neither; $\pi_{77}$: low 3, high 4, Toda 1 — neither; $\pi_{78}$: low 2, high 3, Toda 2 — as low; $\pi_{79}$: low 3, high 2, Toda 1 — neither; $\pi_{80}$: low 2, high 1, Toda 0 — neither; $\pi_{81}$: low 2, high 3, Toda 2 — as low.
All differentials (“low”), $n\le83$
| from (s, c) | to (s, c) | rank | |
|---|---|---|---|
| $d_{2}$ | $\pi_{24}$ (3, 0) | $\pi_{23}$ (5, 1) | 1 |
| $d_{2}$ | $\pi_{25}$ (3, 1) | $\pi_{24}$ (5, 2) | 1 |
| $d_{2}$ | $\pi_{28}$ (4, 1) | $\pi_{27}$ (6, 2) | 1 |
| $d_{2}$ | $\pi_{29}$ (4, 2) | $\pi_{28}$ (6, 3) | 1 |
| $d_{2}$ | $\pi_{32}$ (5, 2) | $\pi_{31}$ (7, 3) | 1 |
| $d_{2}$ | $\pi_{33}$ (5, 3) | $\pi_{32}$ (7, 4) | 1 |
| $d_{2}$ | $\pi_{34}$ (5, 0) | $\pi_{33}$ (7, 1) | 1 |
| $d_{2}$ | $\pi_{35}$ (5, 1) | $\pi_{34}$ (7, 2) | 1 |
| $d_{2}$ | $\pi_{36}$ (6, 3) | $\pi_{35}$ (8, 4) | 1 |
| $d_{2}$ | $\pi_{37}$ (6, 4) | $\pi_{36}$ (8, 5) | 1 |
| $d_{2}$ | $\pi_{38}$ (6, 1) | $\pi_{37}$ (8, 2) | 1 |
| $d_{2}$ | $\pi_{39}$ (6, 2) | $\pi_{38}$ (8, 3) | 1 |
| $d_{2}$ | $\pi_{40}$ (7, 4) | $\pi_{39}$ (9, 5) | 1 |
| $d_{2}$ | $\pi_{41}$ (7, 5) | $\pi_{40}$ (9, 6) | 1 |
| $d_{2}$ | $\pi_{42}$ (7, 2) | $\pi_{41}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{43}$ (7, 3) | $\pi_{42}$ (9, 4) | 1 |
| $d_{2}$ | $\pi_{44}$ (7, 0) | $\pi_{43}$ (9, 1) | 1 |
| $d_{2}$ | $\pi_{44}$ (8, 5) | $\pi_{43}$ (10, 6) | 1 |
| $d_{2}$ | $\pi_{45}$ (7, 1) | $\pi_{44}$ (9, 2) | 1 |
| $d_{2}$ | $\pi_{45}$ (8, 6) | $\pi_{44}$ (10, 7) | 1 |
| $d_{2}$ | $\pi_{46}$ (8, 3) | $\pi_{45}$ (10, 4) | 1 |
| $d_{2}$ | $\pi_{47}$ (8, 4) | $\pi_{46}$ (10, 5) | 1 |
| $d_{2}$ | $\pi_{48}$ (8, 1) | $\pi_{47}$ (10, 2) | 1 |
| $d_{2}$ | $\pi_{48}$ (9, 6) | $\pi_{47}$ (11, 7) | 1 |
| $d_{2}$ | $\pi_{49}$ (8, 2) | $\pi_{48}$ (10, 3) | 1 |
| $d_{2}$ | $\pi_{49}$ (9, 7) | $\pi_{48}$ (11, 8) | 1 |
| $d_{2}$ | $\pi_{50}$ (9, 4) | $\pi_{49}$ (11, 5) | 1 |
| $d_{2}$ | $\pi_{51}$ (9, 5) | $\pi_{50}$ (11, 6) | 1 |
| $d_{2}$ | $\pi_{52}$ (10, 7) | $\pi_{51}$ (12, 8) | 1 |
| $d_{2}$ | $\pi_{52}$ (9, 2) | $\pi_{51}$ (11, 3) | 1 |
| $d_{2}$ | $\pi_{53}$ (9, 3) | $\pi_{52}$ (11, 4) | 1 |
| $d_{2}$ | $\pi_{53}$ (10, 8) | $\pi_{52}$ (12, 9) | 1 |
| $d_{2}$ | $\pi_{54}$ (9, 0) | $\pi_{53}$ (11, 1) | 1 |
| $d_{2}$ | $\pi_{54}$ (10, 5) | $\pi_{53}$ (12, 6) | 1 |
| $d_{2}$ | $\pi_{55}$ (9, 1) | $\pi_{54}$ (11, 2) | 1 |
| $d_{2}$ | $\pi_{55}$ (10, 6) | $\pi_{54}$ (12, 7) | 1 |
| $d_{2}$ | $\pi_{56}$ (10, 3) | $\pi_{55}$ (12, 4) | 1 |
| $d_{2}$ | $\pi_{56}$ (11, 8) | $\pi_{55}$ (13, 9) | 1 |
| $d_{2}$ | $\pi_{57}$ (10, 4) | $\pi_{56}$ (12, 5) | 1 |
| $d_{2}$ | $\pi_{57}$ (11, 9) | $\pi_{56}$ (13, 10) | 1 |
| $d_{2}$ | $\pi_{58}$ (10, 1) | $\pi_{57}$ (12, 2) | 1 |
| $d_{2}$ | $\pi_{58}$ (11, 6) | $\pi_{57}$ (13, 7) | 1 |
| $d_{2}$ | $\pi_{59}$ (10, 2) | $\pi_{58}$ (12, 3) | 1 |
| $d_{2}$ | $\pi_{59}$ (11, 7) | $\pi_{58}$ (13, 8) | 1 |
| $d_{2}$ | $\pi_{60}$ (11, 4) | $\pi_{59}$ (13, 5) | 1 |
| $d_{2}$ | $\pi_{60}$ (12, 9) | $\pi_{59}$ (14, 10) | 1 |
| $d_{2}$ | $\pi_{61}$ (11, 5) | $\pi_{60}$ (13, 6) | 1 |
| $d_{2}$ | $\pi_{61}$ (12, 10) | $\pi_{60}$ (14, 11) | 1 |
| $d_{2}$ | $\pi_{62}$ (12, 7) | $\pi_{61}$ (14, 8) | 1 |
| $d_{2}$ | $\pi_{62}$ (11, 2) | $\pi_{61}$ (13, 3) | 1 |
| $d_{2}$ | $\pi_{63}$ (11, 3) | $\pi_{62}$ (13, 4) | 1 |
| $d_{2}$ | $\pi_{63}$ (12, 8) | $\pi_{62}$ (14, 9) | 1 |
| $d_{2}$ | $\pi_{64}$ (11, 0) | $\pi_{63}$ (13, 1) | 1 |
| $d_{2}$ | $\pi_{64}$ (12, 5) | $\pi_{63}$ (14, 6) | 1 |
| $d_{2}$ | $\pi_{64}$ (13, 10) | $\pi_{63}$ (15, 11) | 1 |
| $d_{2}$ | $\pi_{65}$ (11, 1) | $\pi_{64}$ (13, 2) | 1 |
| $d_{2}$ | $\pi_{65}$ (12, 6) | $\pi_{64}$ (14, 7) | 1 |
| $d_{2}$ | $\pi_{65}$ (13, 11) | $\pi_{64}$ (15, 12) | 1 |
| $d_{2}$ | $\pi_{66}$ (12, 3) | $\pi_{65}$ (14, 4) | 1 |
| $d_{2}$ | $\pi_{66}$ (13, 8) | $\pi_{65}$ (15, 9) | 1 |
| $d_{2}$ | $\pi_{67}$ (12, 4) | $\pi_{66}$ (14, 5) | 1 |
| $d_{2}$ | $\pi_{67}$ (13, 9) | $\pi_{66}$ (15, 10) | 1 |
| $d_{2}$ | $\pi_{68}$ (12, 1) | $\pi_{67}$ (14, 2) | 1 |
| $d_{2}$ | $\pi_{68}$ (13, 6) | $\pi_{67}$ (15, 7) | 1 |
| $d_{2}$ | $\pi_{68}$ (14, 11) | $\pi_{67}$ (16, 12) | 1 |
| $d_{2}$ | $\pi_{69}$ (12, 2) | $\pi_{68}$ (14, 3) | 1 |
| $d_{2}$ | $\pi_{69}$ (13, 7) | $\pi_{68}$ (15, 8) | 1 |
| $d_{2}$ | $\pi_{69}$ (14, 12) | $\pi_{68}$ (16, 13) | 1 |
| $d_{2}$ | $\pi_{70}$ (13, 4) | $\pi_{69}$ (15, 5) | 1 |
| $d_{2}$ | $\pi_{70}$ (14, 9) | $\pi_{69}$ (16, 10) | 1 |
| $d_{2}$ | $\pi_{71}$ (13, 5) | $\pi_{70}$ (15, 6) | 1 |
| $d_{2}$ | $\pi_{71}$ (14, 10) | $\pi_{70}$ (16, 11) | 1 |
| $d_{2}$ | $\pi_{72}$ (6, 3) | $\pi_{71}$ (8, 4) | 1 |
| $d_{2}$ | $\pi_{72}$ (14, 7) | $\pi_{71}$ (16, 8) | 1 |
| $d_{2}$ | $\pi_{72}$ (15, 12) | $\pi_{71}$ (17, 13) | 1 |
| $d_{2}$ | $\pi_{73}$ (6, 0) | $\pi_{72}$ (8, 1) | 1 |
| $d_{2}$ | $\pi_{73}$ (6, 4) | $\pi_{72}$ (8, 5) | 1 |
| $d_{2}$ | $\pi_{72}$ (13, 2) | $\pi_{71}$ (15, 3) | 1 |
| $d_{2}$ | $\pi_{73}$ (13, 3) | $\pi_{72}$ (15, 4) | 1 |
| $d_{2}$ | $\pi_{73}$ (14, 8) | $\pi_{72}$ (16, 9) | 1 |
| $d_{2}$ | $\pi_{73}$ (15, 13) | $\pi_{72}$ (17, 14) | 1 |
| $d_{2}$ | $\pi_{74}$ (13, 0) | $\pi_{73}$ (15, 1) | 1 |
| $d_{2}$ | $\pi_{74}$ (14, 5) | $\pi_{73}$ (16, 6) | 1 |
| $d_{2}$ | $\pi_{74}$ (15, 10) | $\pi_{73}$ (17, 11) | 1 |
| $d_{2}$ | $\pi_{75}$ (13, 1) | $\pi_{74}$ (15, 2) | 1 |
| $d_{2}$ | $\pi_{75}$ (14, 6) | $\pi_{74}$ (16, 7) | 1 |
| $d_{2}$ | $\pi_{75}$ (15, 11) | $\pi_{74}$ (17, 12) | 1 |
| $d_{2}$ | $\pi_{75}$ (6, 2) | $\pi_{74}$ (8, 3) | 1 |
| $d_{2}$ | $\pi_{76}$ (7, 4) | $\pi_{75}$ (9, 5) | 1 |
| $d_{2}$ | $\pi_{76}$ (14, 3) | $\pi_{75}$ (16, 4) | 1 |
| $d_{2}$ | $\pi_{76}$ (15, 8) | $\pi_{75}$ (17, 9) | 1 |
| $d_{2}$ | $\pi_{76}$ (16, 13) | $\pi_{75}$ (18, 14) | 1 |
| $d_{2}$ | $\pi_{77}$ (7, 5) | $\pi_{76}$ (9, 6) | 1 |
| $d_{2}$ | $\pi_{77}$ (14, 4) | $\pi_{76}$ (16, 5) | 1 |
| $d_{2}$ | $\pi_{77}$ (15, 9) | $\pi_{76}$ (17, 10) | 1 |
| $d_{2}$ | $\pi_{77}$ (16, 14) | $\pi_{76}$ (18, 15) | 1 |
| $d_{2}$ | $\pi_{78}$ (14, 1) | $\pi_{77}$ (16, 2) | 1 |
| $d_{2}$ | $\pi_{78}$ (15, 6) | $\pi_{77}$ (17, 7) | 1 |
| $d_{2}$ | $\pi_{78}$ (16, 11) | $\pi_{77}$ (18, 12) | 1 |
| $d_{2}$ | $\pi_{78}$ (7, 2) | $\pi_{77}$ (9, 3) | 1 |
| $d_{2}$ | $\pi_{79}$ (14, 2) | $\pi_{78}$ (16, 3) | 1 |
| $d_{2}$ | $\pi_{79}$ (15, 7) | $\pi_{78}$ (17, 8) | 1 |
| $d_{2}$ | $\pi_{79}$ (16, 12) | $\pi_{78}$ (18, 13) | 1 |
| $d_{2}$ | $\pi_{79}$ (7, 3) | $\pi_{78}$ (9, 4) | 1 |
| $d_{2}$ | $\pi_{80}$ (8, 5) | $\pi_{79}$ (10, 6) | 1 |
| $d_{2}$ | $\pi_{80}$ (15, 4) | $\pi_{79}$ (17, 5) | 1 |
| $d_{2}$ | $\pi_{80}$ (16, 9) | $\pi_{79}$ (18, 10) | 1 |
| $d_{2}$ | $\pi_{80}$ (17, 14) | $\pi_{79}$ (19, 15) | 1 |
| $d_{2}$ | $\pi_{81}$ (8, 6) | $\pi_{80}$ (10, 7) | 1 |
| $d_{2}$ | $\pi_{81}$ (15, 5) | $\pi_{80}$ (17, 6) | 1 |
| $d_{2}$ | $\pi_{81}$ (16, 10) | $\pi_{80}$ (18, 11) | 1 |
| $d_{2}$ | $\pi_{81}$ (17, 15) | $\pi_{80}$ (19, 16) | 1 |
| $d_{2}$ | $\pi_{82}$ (16, 7) | $\pi_{81}$ (18, 8) | 1 |
| $d_{2}$ | $\pi_{82}$ (17, 12) | $\pi_{81}$ (19, 13) | 1 |
| $d_{2}$ | $\pi_{82}$ (8, 3) | $\pi_{81}$ (10, 4) | 1 |
| $d_{2}$ | $\pi_{83}$ (6, 2) | $\pi_{82}$ (8, 3) | 1 |
| $d_{2}$ | $\pi_{83}$ (8, 0) | $\pi_{82}$ (10, 1) | 1 |
| $d_{2}$ | $\pi_{82}$ (15, 2) | $\pi_{81}$ (17, 3) | 1 |
| $d_{2}$ | $\pi_{83}$ (15, 3) | $\pi_{82}$ (17, 4) | 1 |
| $d_{2}$ | $\pi_{83}$ (16, 8) | $\pi_{82}$ (18, 9) | 1 |
| $d_{2}$ | $\pi_{83}$ (17, 13) | $\pi_{82}$ (19, 14) | 1 |
| $d_{2}$ | $\pi_{83}$ (8, 4) | $\pi_{82}$ (10, 5) | 1 |
| $d_{6}$ | $\pi_{61}$ (6, 0) | $\pi_{60}$ (12, 1) | 1 |
| $d_{6}$ | $\pi_{71}$ (8, 0) | $\pi_{70}$ (14, 1) | 1 |
| $d_{6}$ | $\pi_{81}$ (10, 0) | $\pi_{80}$ (16, 1) | 1 |
| $d_{6}$ | $\pi_{82}$ (5, 0) | $\pi_{81}$ (11, 1) | 1 |
The full list up to $n=150$ is in differentials.json.
What the rule is missing
For $n\le82$ the rule disagrees with Toda in $\pi_{49}$, $\pi_{50}$, $\pi_{51}$, $\pi_{59}$, $\pi_{60}$, $\pi_{63}$, $\pi_{64}$, $\pi_{66}$, $\pi_{67}$, $\pi_{68}$, $\pi_{69}$, $\pi_{70}$, $\pi_{73}$, $\pi_{74}$, $\pi_{75}$, $\pi_{76}$, $\pi_{77}$, $\pi_{79}$, $\pi_{80}$. Almost all of the extra classes are removed by arrows $d_5$ that do not change the colour; at $p=3$ this is $d_{2p-1}$, the length of Toda's differential. Allowing such $d_5$ in addition to the rule (with the same greedy filling) gives the following $d_5$:
| from (s, c) | to (s, c) | rank | |
|---|---|---|---|
| $d_{5}$ | $\pi_{50}$ (5, 0) | $\pi_{49}$ (10, 0) | 1 |
| $d_{5}$ | $\pi_{51}$ (5, 1) | $\pi_{50}$ (10, 1) | 1 |
| $d_{5}$ | $\pi_{58}$ (5, 0) | $\pi_{57}$ (10, 0) | 1 |
| $d_{5}$ | $\pi_{60}$ (7, 0) | $\pi_{59}$ (12, 0) | 1 |
| $d_{5}$ | $\pi_{61}$ (7, 1) | $\pi_{60}$ (12, 1) | 1 |
| $d_{5}$ | $\pi_{68}$ (7, 0) | $\pi_{67}$ (12, 0) | 1 |
| $d_{5}$ | $\pi_{70}$ (9, 0) | $\pi_{69}$ (14, 0) | 1 |
| $d_{5}$ | $\pi_{71}$ (9, 1) | $\pi_{70}$ (14, 1) | 1 |
| $d_{5}$ | $\pi_{74}$ (6, 1) | $\pi_{73}$ (11, 1) | 1 |
| $d_{5}$ | $\pi_{76}$ (7, 0) | $\pi_{75}$ (12, 0) | 1 |
| $d_{5}$ | $\pi_{77}$ (7, 1) | $\pi_{76}$ (12, 1) | 1 |
| $d_{5}$ | $\pi_{78}$ (9, 0) | $\pi_{77}$ (14, 0) | 1 |
| $d_{5}$ | $\pi_{80}$ (11, 0) | $\pi_{79}$ (16, 0) | 1 |
| $d_{5}$ | $\pi_{82}$ (5, 0) | $\pi_{81}$ (10, 0) | 1 |
| $d_{5}$ | $\pi_{81}$ (11, 1) | $\pi_{80}$ (16, 1) | 1 |
Disagreements remain in $\pi_{57}$, $\pi_{58}$, $\pi_{63}$, $\pi_{64}$, $\pi_{66}$, $\pi_{67}$, $\pi_{73}$, $\pi_{74}$, $\pi_{75}$, $\pi_{76}$, $\pi_{78}$, $\pi_{79}$, $\pi_{80}$: there the greedy $d_5$ kills classes that are alive in Toda. So Toda needs some colour-preserving $d_5$, but not all the allowed ones. None of the purely colour-based variants we checked reproduces Toda beyond $\pi_{48}$.