Enumeration of finite automata

Question

There is a nice paper Enumeration of Finite Automata by Frank Harary and Ed Palmer which presents a formula $a(n,k,m)$ for the number of finite automata with $n$ states, $k$ input symbols and $m$ output symbols. It is stated in Corollary $3$ as \begin{align*} a(n,k,m)=\frac{1}{n!k!m!}\sum_{H_1} I(\alpha,\beta,\alpha)I(\alpha,\beta,\gamma) \end{align*} where the sum is over all permutations in $H_1=S_n^{S_n\times S_k}\times S_m^{S_n\times S_k}$ of the form $\{[(\alpha,\beta);\alpha^{-1}],[(\alpha,\beta);\gamma]\}$ with \begin{align*} I(\alpha,\beta,\gamma)=\prod_{p=1}^n\prod_{q=1}^k\left[\sum_{s|[p,q]}sj_s(\gamma)\right]^{j_p(\alpha)j_q(\beta)\langle p,q\rangle} \end{align*}

Here the authors denote with $[p,q]:=\operatorname{lcm}(p,q), \langle p,q\rangle:=\operatorname{gcd}(p,q)$ and $j_p(\alpha)$ denotes the number of cycles of the permutation $\alpha$ with length $p$.

The special case $n=k,m=1$ is already analysed and calculated for small values of $n$ in this MSE post, especially with focus to $n=4$.

Question: Maybe someone could provide some computations for the general case for small values of $n,k,m$?

Answer 1

In answering this question we refer to the algorithm at the MSE link which works for the generalized problem as well. The only difference is that the values that go into the slots of the array/table are pairs of states and output symbols, meaning when we transition from a certain column on an input symbol corresponding to a row we transition to the state (first element of the pair) and output the symbol (second element of the pair). The action on the slots is the simultaneous action of $\pi$ and $\tau$ on the rows and columns and we now have a permutation $\sigma$ which acts on the set of output symbols and the action on the values is the combined action of $\tau$ and $\sigma$ on the state / symbol pairs.

We get the following table for one output symbol.

|   1|          1|                 1|                       1|                              1|                                    1|                                           1|                                                 1|
|   3|          7|                13|                      22|                             34|                                   50|                                          70|                                                95|
|   7|         74|               638|                    4663|                          28529|                               151600|                                      713176|                                           3028727|
|  19|       1474|            118949|                 7643021|                      396979499|                          17265522590|                                646203233957|                                    21243806443115|
|  47|      41876|          42483668|             33179970333|                 20762461502595|                    10831034126757463|                         4844565331763027596|                            1896647286212566394157|
| 130|    1540696|       23524514635|         274252613077267|            2559276179593762172|              19903050866658120066632|                 132673733865643566661223817|                    773869304738817313660236854435|
| 343|   68343112|    18477841853059|     3802866637652928476|       626361440405926396941497|        85973094952794304259466151418|         10114722264843500593900485682759058|          1041247439945746392774732251877428013424|
| 951| 3540691525| 19526400231564564| 81874932562648494674439| 274724907231470170012527305235| 768186632385442429091738459545921683| 1841148232300929744056375072663778725072045| 3861169308385212945415179151162048048461447621051|

For two output symbols we have

|     1|               2|                        2|                                 3|                                          3|                                                   4|                                                           4|                                                                    5|
|     6|              44|                      226|                              1036|                                       4006|                                               13876|                                                       43186|                                                               123706|
|    22|            2038|                   142336|                           7775708|                                  341906882|                                         12592855970|                                                399366367444|                                                       11132314379998|
|   114|          176936|                238882846|                      244698934716|                            200649261017386|                                  137143648460408272|                                        80366174079209158078|                                              41217801421317353953038|
|   538|        20943790|             694540531869|                 17362195783419565|                      347256965617453111707|                           5787905149678353796143590|                               82689320232608432438262174088|                                   1033688856029644143398545746261666|
|  2800|      3108818680|         3081614657394158|            2300263170022800838590|               1373710145403734491538076692|                   683647218221456315461840833799588|                     291623393789554111334921119339297251576|                        108848103655093534827120896470552784018126133|
| 14435|    553255960308|     19368605578168164179|       510403370619400317035233276|        10760675018954199971112474584547034|          189053417206572805331242303827478007687534|           2846969183281612697167894035560332610102537605107|             37513627164757945129191686915360296965220882487348368322|
| 76312| 114776687721990| 163754994767359896315206| 175823884588034784365611422263567| 151031502945525188132621372232074129315388| 108112560585492844973667875651850996929528575835574| 66334273232261168899346826889209523621370385072001650536116| 35612941825082950044316879351953518880328546726186269125209259942000|

Three output symbols yield

|      1|                 2|                           3|                                      4|                                                5|                                                           7|                                                                     8|                                                                              10|
|      6|                74|                         775|                                   7124|                                            55668|                                                      377269|                                                               2255068|                                                                        12102178|
|     29|              7623|                     1804128|                              329641077|                                      48317584819|                                               5910777204447|                                                       620630699132987|                                                               57098016161377374|
|    190|           1501516|                 10322146155|                         53512221536494|                               221968136483832014|                                       767306804276224740828|                                             2273639672252875423729778|                                                    5895263464882668948056075498|
|   1289|         401371270|             101367856946674|                   19243544529701850104|                        2922627429145967591227933|                              369897467120287921148106491100|                                   40127586921742103692252419866530400|                                        3809020901470314640315364328599642887506|
|   9673|      134138227473|         1518024410618449355|             12907594258334064169919121|                 87803188849193004851359368791756|                     497730359833453928180319002991414602093|                        2418417068028280199534213597754694851805840225|                           10281969996512134071147543063509604282591387558257520|
|  73604|    53725010241266|     32201676604966459555889|       14499308203534486200843433873288|         5222906915046943511008193569385565417541|           1567819635143439097415728431946215896270059293161|             403397426941463986598664115278880491308873007636372427413|               90819310744609116970288225981171645606992548661728301980002516662|
| 573442| 25081227120200634| 918865057207831149035535828| 25285803348327743049043999665003927370| 556668502782671968664754976635618690023788914186| 10212576716712592462402577334011641314012112279662417473469| 160593227242102911238351158110065456181421151497935704882980552606514| 2209668743041973325985756217800328983151637526070225333484395817216844313778044|

Four symbols yield

|       1|                  2|                              3|                                         5|                                                     6|                                                                9|                                                                          11|                                                                                     15|
|       6|                 81|                           1183|                                     17320|                                                223743|                                                          2527953|                                                                    25100642|                                                                              222144431|
|      29|              11676|                        6064606|                                2593640209|                                          897009602752|                                                  259029607981273|                                                           64163314527895517|                                                                   13915354324987224434|
|     209|            3831148|                    81573276196|                          1334647986999812|                                  17493019379544106141|                                         191083931326433751661244|                                                1789145512052234025354299479|                                                       14658245204843197745963032946030|
|    1605|         1790644262|               1896670209705424|                    1517048789183286280242|                           970906913413864886205472630|                                517817738821504564293534451239523|                                     236717123156531446119639354041331039161|                                           94687056373953999303903668799913187496263156|
|   14581|      1059379897194|           67316410303471722434|              3215992447007150335848738654|                  122917096383192644964591012637376201|                      3914970565374711299589044295533654728633307|                         106880364202506644619748019682746095700393900152769|                             2553144552899651934745530164746582340956543973128820263056|
|  139393|    753537775187942|      3384772964731425916075399|       11417522742490309099171117430032545|          30811161705715253062014503052903675566658545|            69288800372821565423720577304077855202305583626701885|             133558404360787903168869516536280931557107488047811301767090944|               225261750393971075099732774525570356293879964632402213718679044305894097|
| 1396571| 625251791124395555| 228938436067723951049495991006| 62929794221715160999635636523327894882612| 13838407708142508413727196626725814975774777251143465| 2535915030456565177161444959970701001632828430350354446662490458| 398324009007397248996962807526047717969141597988514641606768498289421689877| 54745234941096457415294245370001308972451724232455240696557887565208148810995582605398|

Five yield

|       1|                   2|                               3|                                            5|                                                        7|                                                                   10|                                                                               13|                                                                                           18|
|       6|                  81|                            1283|                                        23718|                                                   427097|                                                              7038183|                                                                        103821898|                                                                                   1372476565|
|      29|               12621|                         9875766|                                   7694431189|                                            5108729338005|                                                     2866744631627614|                                                              1383444387175373624|                                                                        584738631310521555854|
|     209|             5269634|                    242293771832|                             9508729532667775|                                    303537782294910006324|                                            8092008307288214998320242|                                                   184959457244832433282602143175|                                                           3699331066099122391214267900044654|
|    1652|          3522483774|               10830193709142911|                      26326043763404282897041|                            51400728418762283743166947873|                                  83657888529920202329649049898106090|                                       116710057646947398301738658574346204631684|                                             142468411615177769332030145694979476640229799189|
|   15851|       3145805694347|           748102205731495912974|               136208975222504119847429651282|                   19858370962230255015514418124978318079|                       2412787002750586428934439397030434799264061139|                           251274509502830170033287481345380174207693056359521218|                               22897389793260955643229220128574252798224672181928261140465132|
|  166704|    3451400880452119|      73411241836287162439679965|        1180551376563438246848941889675139885|          15191078168438817387019547141066538853359987716|            162897187467367310343607416594982652886027395559664704867|              1497241493787657622590696899117249253525915361372369634716838093562|                12041433120029892610323311791551075975557111745695889862014376128073109032711|
| 1903565| 4453493876743114141| 9696353154834682640039652745383| 15885725788645815939897203091966549890622620| 20821721985157272922019024288021084022442430164588852951| 22742872566990523157952024381064067346859577618809430167348089260678| 21292527088890116346521008056915214793265230093299707297080802600716512185635775| 17442838191172723332310678848004599133452884005515399679140805741625547114446168989072880538|

Finally we get for six symbols

|       1|                    2|                                 3|                                             5|                                                          7|                                                                      11|                                                                                  14|                                                                                               20|
|       6|                   81|                              1296|                                         25462|                                                     538398|                                                                11293138|                                                                           222523395|                                                                                       4028465835|
|      29|                12695|                          11328242|                                   12588216476|                                             13507531099557|                                                       12816676023294742|                                                                10610435880654869474|                                                                           7727294095780485593467|
|     209|              5635034|                      396518228841|                             29902254119865429|                                     1947536351902062154396|                                             107300432454566001311927042|                                                     5082116300041019725568491696927|                                                             210740137620013511032529954013222997|
|    1652|           4452248665|                 28661573376513712|                      168916250895768873373125|                             817701868164546859278494745163|                                   3309982213851389919369842502624515185|                                        11489588802579132510260340618793545674029229|                                               34899323818332948931809633587657800749959429140381|
|   15981|        5147747713851|            3350282292788028229116|               1806224092274722460193800299488|                    785710893213334594665452752490935409600|                        285033249600409431428643990739291312182972084132|                            88635922473731155883430561365483225722614035385062387241|                                24117625609927898779221726509298149056270088412428821435055351917|
|  171494|     7721337186134447|       564461055370558962491069562|        32440112974696247296439224174402635608|          1495496356773389913366753876131348301821086183629|             57461231472727120738649283370058285613319924784137652332510|              1892438067444572851650149500498661434054764424790064535313952779756847|                 54535174475104423211660022224834911399436311980199332329136348183225443895408945|
| 2041940| 14003166710753529537| 128580139323392617149472430498611| 905045555050578843422814928359489284108944076| 5100536012710000997786910449314715054126988193344281363091| 23954874543703392448557828429937283387539096055257084732285733625757877| 96433013296267950226465899688337115485213562485757881838921103460239405763926814829| 339676614862729029614552301296020122485910436927008569295805654935518977116532247635480871741432|

The Maple code for this was as follows.

with(combinat);

pet_cycleind_symm :=
proc(n)
local p, s;
option remember;

    if n=0 then return 1; fi;

    expand(1/n*add(a[l]*pet_cycleind_symm(n-l), l=1..n));
end;

pet_flatten_term :=
proc(varp)
local terml, d, cf, v;

    terml := [];

    cf := varp;
    for v in indets(varp) do
        d := degree(varp, v);
        terml := [op(terml), seq(v, k=1..d)];
        cf := cf/v^d;
    od;

    [cf, terml];
end;

cycles_prod :=
proc(cyca, cycb)
local ca, cb, lena, lenb, res, vlcm;

    res := 1;

    for ca in cyca do
        lena := op(1, ca);

        for cb in cycb do
            lenb := op(1, cb);

            vlcm := lcm(lena, lenb);
            res := res*a[vlcm]^(lena*lenb/vlcm);
        od;
    od;

    res;
end;


automaton :=
proc(N, M, K)
option remember;
local idx_slots, idx_cols, idx_syms, res, a, b, c, sim, flat_sim,
    sym, flat_sym, flat_a, flat_b, flat_c,
    cyc_a, cyc_b, len_a, len_b, p, q;

    if N > 1 then
        idx_slots := pet_cycleind_symm(N);
    else
        idx_slots := [a[1]];
    fi;

    if M > 1 then
        idx_cols := pet_cycleind_symm(M);
    else
        idx_cols := [a[1]];
    fi;

    if K > 1 then
        idx_syms := pet_cycleind_symm(K);
    else
        idx_syms := [a[1]];
    fi;


    res := 0;

    for a in idx_slots do
        flat_a := pet_flatten_term(a);
        for b in idx_cols do
            flat_b := pet_flatten_term(b);

            sim := cycles_prod(flat_a[2], flat_b[2]);
            flat_sim := pet_flatten_term(sim);

            for c in idx_syms do
                flat_c := pet_flatten_term(c);

                sym := cycles_prod(flat_b[2], flat_c[2]);
                flat_sym := pet_flatten_term(sym);

                p := 1;

                for cyc_a in flat_sim[2] do
                    len_a := op(1, cyc_a);
                    q := 0;

                    for cyc_b in flat_sym[2] do
                        len_b := op(1, cyc_b);

                        if len_a mod len_b = 0 then
                            q := q + len_b;
                        fi;
                    od;

                    p := p*q;
                od;

                res := res +
                p*flat_a[1]*flat_b[1]*flat_c[1];
            od;
        od;
    od;

    res;
end;


output :=
proc(MXN, MXM, K)
local data, N, M, fd, fname, width;

    data := table();

    for N to MXN do
        data[N] := table();

        for M to MXM do
            data[N][M] := automaton(M, N, K);
        od;
    od;

    fname := sprintf("automata-%d-%d-%d.txt", MXN, MXM, K);
    fd := fopen(fname, WRITE);

    for N to MXN do
        fprintf(fd, "|");
        for M to MXM do
            width := nops(convert(data[MXN][M], base, 10));
            fprintf(fd, "% *d|", width+1, data[N][M]);
        od;
        fprintf(fd, "\n");
    od;

    fclose(fd);
end;

Answer 2

Below are tables for $a(N,K,M)$ for values $\le 5$ where

$N = $ number of states, $K = $ number of inputs, $M = $ number of outputs.

Note that if we fix $N$ and $K$ and let $M$ increase, this eventually becomes independent of $M$. Once $M$ is sufficiently large to cover any possible equivalence over the edges of the state transtion graph, there is no gain from increasing it.

So two interesting questions:

What is the terminal $M$ for each choice of $(N,K)$? I.e. calculate $$ M^*(N,K) := m^*: a(N,K,m) = a(N,K,m^*)\ \ \ \forall \ m>m^* $$ What are the values of $a(N,K,M^*(N,K))$?

+-----+---+------+------------+-----------------+-----------------+-----------------+
| M=2 |   | K    |            |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     |   | 1    | 2          | 3               | 4               | 5               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
| N   | 1 | 1    | 2          | 2               | 3               | 3               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 2 | 6    | 44         | 226             | 1036            | 4006            |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 3 | 22   | 2038       | 142336          | 7775708         | 341906882       |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 4 | 114  | 176936     | 238882846       | 244698934716    | 200649261017386 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 5 | 538  | 20943790   | 694540531869    | 1.35E+015       |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     |   |      |            |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
| M=3 |   | K    |            |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     |   | 1    | 2          | 3               | 4               | 5               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
| N   | 1 | 1    | 2          | 3               | 4               | 5               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 2 | 6    | 75         | 775             | 7124            | 55668           |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 3 | 29   | 7623       | 1804128         | 329641077       | 48317584819     |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 4 | 190  | 1501516    | 10322146155     | 53512221536494  |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 5 | 1289 | 401371270  | 101367856946674 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     |   |      |            |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
| M=4 |   | K    |            |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     |   | 1    | 2          | 3               | 4               | 5               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
| N   | 1 | 1    | 2          | 3               | 5               | 6               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 2 | 6    | 81         | 1183            | 17320           | 223743          |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 3 | 29   | 11676      | 6064606         | 2593640209      | 897009602752    |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 4 | 209  | 3831148    | 81573276196     | 248097408553    |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 5 | 1605 | 1790644262 |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     |   |      |            |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
| M=5 |   | K    |            |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     |   | 1    | 2          | 3               | 4               | 5               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
| N   | 1 | 1    | 2          | 3               | 5               | 7               |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 2 | 6    | 81         | 1283            | 23718           | 427097          |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 3 | 29   | 12621      | 9875766         | 7694431189      | 5108729338005   |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 4 | 209  | 5269634    | 242293771832    | 167930305528968 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+
|     | 5 | 1652 | 3522483774 |                 |                 |                 |
+-----+---+------+------------+-----------------+-----------------+-----------------+