The following creates the numbers a_k, b’_k and b_k that Morita uses.

Clear[a];
a[k_]:=Factorial2[6k-1];
bprime[K_]:=Coefficient[Log[Sum[Factorial2[(6k-1)]/(Factorial2[2k-1])x^k/k!,{k,0,K+1}]+O[x]^(K+1)],x,K];
b[K_]:=Factorial2[2K-1]K!bprime[K];

The following commutes the Morita' s c_k^{(i)} as CC[k, i]. These are recurvisely determined by a sum that has three terms (as well as setting some boundary cases to be  0 by hand).

Clear[CC];
CC[0,1]=0;
CC[k_,i_]:=CC[k,i]=If[i==0 ∨ i>2k,0, Factorial2[6k-2i-1]-(
STerm[k,i]+
I0Term[k,i]
)]

The S-term is a sum over previous numbers:

STerm[k_,i_]:=Sum[Binomial[2k-i,2s]CC[k-s,i]a[s],{s,1,k-i+1}]

The  i0-term is a sum over terms which themselves are sums over partitions:

I0Term[k_,i_]:=Sum[PartitionSum[i,i0,k],{i0,0,Floor[i/2]}]

The coefficient of this sum of partitions is different when i_0 = 0:

PartitionSum[i_,i0_,k_]:=If[i0==0,PartitionSumBare[k,i,i0],Binomial[i,2i0]Factorial2[2i0-1]PartitionSumBare[k,i,i0]];

To compute PartitionSumBare we need to produce the partitions. The partition into a single set is dropped when i0 = 0:

Partitions[i_,i0_]:=Drop[Tally/@IntegerPartitions[i-2i0],1-Min[i0,1]]

For each partition there will be a sum over certain variables l_u^t. These are created by the following function, which outputs a list of (i,j,l).

VariablesCreator[partition_]:=Flatten/@Flatten[Transpose[{ConstantArray[#,Last[#]],Range[1,Last[#]]}]&/@partition,1]

These variables can take certain values. For each variables, this list is created by the following function. It assigns a list (i,j,l,value of l) to a list (i,j,l).

Individualvariables[k_,i0_,variable_]:=Append[variable,#]&/@(Range[Ceiling[variable[[1]]/2+1/4],k-i0])

The following combines these values and throws out those that don’t satisfy k-i0-sum l \geq 0. The result is a list of lists, each of values contains a collection of variables and their value.

VariableValues[k_,i0_,variablelist_]:=Select[Tuples[Individualvariables[k,i0,#]&/@variablelist],k-i0-Total[Transpose[#][[4]]]>-1&]

Given a tuple (i,j,l,value of l), this produces the numbers 1/(2l-i) c^i_l.

Cterm[variablevalue_]:=If[Length[variablevalue]<4,0,1/((2variablevalue[[4]]-variablevalue[[1]])!)CC[variablevalue[[4]],variablevalue[[1]]]]

All these numbers are multiplied, together with a factor a[k-i0-sum l]/(2(k-i0-sum l)):

Cproduct[k_,i0_,variablevalues_] :=Times@@Append[(Cterm/@variablevalues),a[k-i0-Total[Transpose[variablevalues][[4]]]]/((2(k-i0-Total[Transpose[variablevalues][[4]]]))!)]

The following combines all of this: it creates the variables, their values, takes their product and multiplies this with a coefficient.

Coeff[partition2_]:=Times@@(1/(Last[#]!^First[#](First[#]!))&/@partition2)
PartitionTerm[k_,i_,i0_,partition_]:=Coeff[partition]Cproduct[k,i0,#]&/@VariableValues[k,i0,VariablesCreator[partition]];

Finally we sums over all partitions and add an exceptional case i=2i0.

PartitionSumBare[k_,i_,i0_]:=If[(i==2)∧i0==1,0,
 If[i==2i0,a[k-i0],(2k-i)!(i-2i0)!Total[Flatten[PartitionTerm[k,i,i0,#]&/@Partitions[i,i0]]]]]

This allows for a recursive computation of the c[k,i]. Compare to Morita Lemma 6.3, 7.2, 7.7 (c_k = C[k,1], d_2 = C[k,2], f_k = C[k,3], and Example 8.7 saying c[3,4] = 216.

Table[CC[k,i],{k,1,5},{i,1,5}]//TableForm

Now we need to similarly implement the constructions of the relations, by replacing c[l,j] and a[j] by c[j,l]e[l] and alpha[l], as well as multiplying each term corresponding to i0 by e_1^{i0}. Here is alpha:

alpha[l_]:=Factorial2[2l-1]l!Coefficient[Exp[Sum[bprime[k]e[k]t^k,{k,1,l}]+O[t]^(l+1)],t,l]//Expand

Clear[GammaC];
GammaC[0,1]=0;
GammaC[k_,i_]:=GammaC[k,i]=If[i==0 ∨ i>2k,0,
CC[k,i]e[k]+GammaSTerm[k,i]+GammaI0Term[k,i]
]
GammaSTerm[k_,i_]:=Sum[Binomial[2k-i,2s]CC[k-s,i]e[k-s]alpha[s],{s,1,k-i+1}];
GammaI0Term[k_,i_]:=Sum[e[1]^i0 GammaPartitionSum[i,i0,k],{i0,0,Floor[i/2]}];
GammaPartitionSum[i_,i0_,k_]:=If[i0==0,GammaPartitionSumBare[k,i,i0],Binomial[i,2i0]Factorial2[2i0-1]GammaPartitionSumBare[k,i,i0]];

GCterm[variablevalue_]:=If[Length[variablevalue]<4,0,1/((2variablevalue[[4]]-variablevalue[[1]])!)e[variablevalue[[4]]]CC[variablevalue[[4]],variablevalue[[1]]]]
GCproduct[k_,i0_,variablevalues_] :=Times@@Append[(GCterm/@variablevalues),alpha[k-i0-Total[Transpose[variablevalues][[4]]]]/((2(k-i0-Total[Transpose[variablevalues][[4]]]))!)]

GammaPartitionSumBare[k_,i_,i0_]:=If[(i==2)∧i0==1,0,
 If[i==2i0,alpha[k-i0],(2k-i)!(i-2i0)!Total[Flatten[GammaPartitionTerm[k,i,i0,#]&/@Partitions[i,i0]]]]]
GammaPartitionTerm[k_,i_,i0_,partition_]:=Coeff[partition]GCproduct[k,i0,#]&/@VariableValues[k,i0,VariablesCreator[partition]];

The gamma i=1, delta i=2, fourth relation i=3.

Table[GammaC[k,i],{k,1,3},{i,1,3}]//Expand//TableForm

This is Morita’s relation in H^{2k}(M_{3k-i-1}). The alpha are i=0, the gamma i=1, delta i=2, fourth relation i=3. For the sake of the reader, we add the degree and genus, that is, the output is {d,g,relation}. These exist for 0 \leq i \leq 3k-3.

Rel[k_,i_]:=If[i<0∨i>3k-3,"Error",{2k,3k-i-1,alpha[k]+Sum[(3^j i!)/((i-j)!(3k-i-2)^j) Binomial[2k,j]GammaC[k,j],{j,1,i}]}]//Expand
RelBare[k_,i_]:=If[i<0∨i>3k-3,"Error",alpha[k]+Sum[(3^j i!)/((i-j)!(3k-i-2)^j) Binomial[2k,j]GammaC[k,j],{j,1,i}]]//Expand

Compare Morita Definition 5.10:

Table[RelBare[k,0],{k,1,4}]//Expand//TableForm

He usually make the GCD of coefficients equal to 1. These are Examples 6.6, 7.5, 7.10 (note that Morita didn’t normalize the last equation in Example 7.5 for some reason, we’ve added it for comparison).

GCDList[{list___}]:=GCD[list]
DividebyGCD[p_]:=(p/GCDList[Flatten[CoefficientList[p,{e[1],e[2],e[3],e[4],e[5],e[6],e[7],e[8],e[9],e[10],e[11]}]]])

Table[DividebyGCD[RelBare[k,1]],{k,2,4}]//Expand//TableForm
Table[DividebyGCD[RelBare[k,2]],{k,3,4}]//Expand//TableForm
DividebyGCD[3007814342400e[4]+174574650600e[1 ]e[3]+41168962800e[2]^2+7421803200e[1]^2e[2]+108226125e[1]^4]//Expand
Table[DividebyGCD[RelBare[k,3]],{k,3,4}]//Expand//TableForm

DividebyGCD[RelBare[3,3]]//Expand

Here we compute the claim in the proof of Lemma 6.6 of “E2-cells and mapping class groups”

{Rel[7,2][[1]],Rel[7,2][[2]],Simplify[(Rel[7,2][[3]]/.{e[3]->0,e[4]->0,e[5]->0,e[6]->0,e[7]->0})289/184712653125]}

Output for fixed degree 2k or a subset of relations.

AllDeg[k_]:=Table[Rel[k,i],{i,0,3k-3}]//TableForm
SomeDeg[k_,j_]:=Table[Rel[k,i],{i,0,j}]//TableForm

Output for fixed genus g, and degree \leq 2d.

AllGenus[g_,d_]:=Table[Rel[i,3i-g-1],{i,Ceiling[(g+1)/3],d}]

Let’s do genus 24.

AllGenus[24,10]//TableForm

Morita2410=DividebyGCD[AllGenus[24,10][[2,3]]]//Expand

AllGenus[24,11];

Let’s check that the Faber-Zagier relations in genus 24, degree 18 contains the Morita relation:

Morita249=AllGenus[24,10][[1,3]];

FZ249={-63973152000000*k1^9+30503170252800000*k1^7*k2-4226875940044800000*k1^5*k2^2-2915954463744000000*k1^6*k3+185219448546263040000*k1^3*k2^3+575288684839895040000*k1^4*k2*k3+357695490869821440000*k1^5*k4-1818374601949839360000*k1*k2^4-22604209586302353408000*k1^2*k2^2*k3-15609532507427045376000*k1^3*k3^2-46878079906152972288000*k1^3*k2*k4-48397703914552688640000*k1^4*k5+98229933579574444032000*k2^3*k3+610878222688381304832000*k1*k2*k3^2+917355610307605561344000*k1*k2^2*k4+1901459467425679618867200*k1^2*k3*k4+3791006982078109581312000*k1^2*k2*k5+6498279927300180934656000*k1^3*k6-2741022667754682974208000*k3^3-24698952328060976549068800*k2*k3*k4-38464415215349056457932800*k1*k4^2-24623945434210982952960000*k2^2*k5-102137077428430669715865600*k1*k3*k5-253442768229738567696384000*k1*k2*k6-779727081853091569257676800*k1^2*k7+2058019270454047574222438400*k4*k5+3400494418229709184067174400*k3*k6+10090780490119840909177651200*k2*k7+72299260937371663092783513600*k1*k8-3807523278753269527616893747200*k9,-194400000000/7*k1^9+100776960000000/7*k1^7*k2-2176782336000000*k1^5*k2^2-1484780544000000*k1^6*k3+104485552128000000*k1^3*k2^3+320712597504000000*k1^4*k2*k3+196781123174400000*k1^5*k4-1128443962982400000*k1*k2^4-13854784212172800000*k1^2*k2^2*k3-9450331206451200000*k1^3*k3^2-28336481737113600000*k1^3*k2*k4-28846719516672000000*k1^4*k5+66502964218429440000*k2^3*k3+408254308118691840000*k1*k2*k3^2+612068005521653760000*k1*k2^2*k4+1252472492780421120000*k1^2*k3*k4+2492356566240460800000*k1^2*k2*k5+4210182631666483200000*k1^3*k6-2004982268760686592000*k3^3-18035603896038064128000*k2*k3*k4-27665473849578749952000*k1*k4^2-17944967276931317760000*k2^2*k5-73441440151885578240000*k1*k3*k5-181879889687992074240000*k1*k2*k6-3857868625971652853760000/7*k1^2*k7+1622225041834591125504000*k4*k5+2679697041403083227136000*k3*k6+55553308213991801094144000/7*k2*k7+55969959696596559986688000*k1*k8-3242177782703275062067200000*k9}/.{k1->(-1)^(1+1)e[1],k2->(-1)^(1+2)e[2],k3->(-1)^(1+3)e[3],k4->(-1)^(1+4)e[4],k5->(-1)^(1+5)e[5],k6->(-1)^(1+6)e[6],k7->(-1)^(1+7)e[7],k8->(-1)^(1+8)e[8],k9->(-1)^(1+9)e[9]};

sol=Solve[{a Coefficient[FZ249[[1]],e[9]]+b Coefficient[FZ249[[2]],e[9]]==Coefficient[Morita249,e[9]],a Coefficient[FZ249[[1]],e[1]e[8]]+b Coefficient[FZ249[[2]],e[1]e[8]]==Coefficient[Morita249,e[1]e[8]]},{a,b}]

Morita249-a FZ249[[1]]-b FZ249[[2]]/.sol//Expand

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system2410=(a1 Coefficient[FZ2410[[1]],#]+a2 Coefficient[FZ2410[[2]],#]+a3 Coefficient[FZ2410[[3]],#]+a4 Coefficient[FZ2410[[4]],#]+a5 Coefficient[FZ2410[[5]],#]+a6 Coefficient[FZ2410[[6]],#]+a7 Coefficient[e[1]FZ249,#]==Coefficient[Morita2410,#]&)/@{e[10],e[1]e[9],e[2]e[8],e[3][e7],e[4]e[6],e[1]^2e[8],e[5]^2};
sol2410=Solve[system2410,{a1,a2,a3,a4,a5,a6,a7}]

Morita2410-a1 FZ2410[[1]]-a2 FZ2410[[2]]-a3 FZ2410[[3]]-a4 FZ2410[[4]]-a5 FZ2410[[5]]-a6 FZ2410[[6]]/.sol2410//FullSimplify