Simple Hurwitz numbers $H_{g,\mu}$

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Numbers

$g$

$\mu$

$H_{g,\mu}$

1, 1, 1:

2, 1, 1:

240

1, 1, 1, 1:

2880

3, 1, 1:

3240

2, 2, 1:

2880

2, 1, 1, 1:

50400

1, 1, 1, 1, 1:

1008000

4, 1, 1:

53760

3, 2, 1:

45360

3, 1, 1, 1:

1088640

2, 2, 2:

40320

2, 2, 1, 1:

967680

2, 1, 1, 1, 1:

26127360

1, 1, 1, 1, 1, 1:

783820800

5, 1, 1:

1050000

4, 2, 1:

860160

4, 1, 1, 1:

27095040

3, 3, 1:

816480

3, 2, 2:

725760

3, 2, 1, 1:

22861440

3, 1, 1, 1, 1:

800150400

2, 2, 2, 1:

20321280

2, 2, 1, 1, 1:

711244800

2, 1, 1, 1, 1, 1:

27382924800

1, 1, 1, 1, 1, 1, 1:

1150082841600

6, 1, 1:

23514624

5, 2, 1:

18900000

5, 1, 1, 1:

756000000

4, 3, 1:

17418240

4, 2, 2:

15482880

4, 2, 1, 1:

619315200

4, 1, 1, 1, 1:

27249868800

3, 3, 2:

14696640

3, 3, 1, 1:

587865600

3, 2, 2, 1:

522547200

3, 2, 1, 1, 1:

22992076800

3, 1, 1, 1, 1, 1:

1103619686400

2, 2, 2, 2:

464486400

2, 2, 2, 1, 1:

20437401600

2, 2, 1, 1, 1, 1:

980995276800

2, 1, 1, 1, 1, 1, 1:

51011754393600

1, 1, 1, 1, 1, 1, 1, 1:

2856658246041600

7, 1, 1:

592950960

6, 2, 1:

470292480

6, 1, 1, 1:

23279477760

5, 3, 1:

425250000

5, 2, 2:

378000000

5, 2, 1, 1:

18711000000

5, 1, 1, 1, 1:

1010394000000

4, 4, 1:

412876800

4, 3, 2:

348364800

4, 3, 1, 1:

17244057600

4, 2, 2, 1:

15328051200

4, 2, 1, 1, 1:

827714764800

4, 1, 1, 1, 1, 1:

48421313740800

3, 3, 3:

330674400

3, 3, 2, 1:

14549673600

3, 3, 1, 1, 1:

785682374400

3, 2, 2, 2:

12933043200

3, 2, 2, 1, 1:

698384332800

3, 2, 1, 1, 1, 1:

40855483468800

3, 1, 1, 1, 1, 1, 1:

2573895458534400

2, 2, 2, 2, 1:

620786073600

2, 2, 2, 1, 1, 1:

36315985305600

2, 2, 1, 1, 1, 1, 1:

2287907074252800

2, 1, 1, 1, 1, 1, 1, 1:

154433727512064000

1, 1, 1, 1, 1, 1, 1, 1, 1:

11119228380868608000

8, 1, 1:

16609443840

7, 2, 1:

13044921120

7, 1, 1, 1:

782695267200

6, 3, 1:

11639738880

6, 2, 2:

10346434560

6, 2, 1, 1:

620786073600

6, 1, 1, 1, 1:

40351094784000

5, 4, 1:

11088000000

5, 3, 2:

9355500000

5, 3, 1, 1:

561330000000

5, 2, 2, 1:

498960000000

5, 2, 1, 1, 1:

32432400000000

5, 1, 1, 1, 1, 1:

2270268000000000

4, 4, 2:

9083289600

4, 4, 1, 1:

544997376000

4, 3, 3:

8622028800

4, 3, 2, 1:

459841536000

4, 3, 1, 1, 1:

29889699840000

4, 2, 2, 2:

408748032000

4, 2, 2, 1, 1:

26568622080000

4, 2, 1, 1, 1, 1:

1859803545600000

4, 1, 1, 1, 1, 1, 1:

139485265920000000

3, 3, 3, 1:

436490208000

3, 3, 2, 2:

387991296000

3, 3, 2, 1, 1:

25219434240000

3, 3, 1, 1, 1, 1:

1765360396800000

3, 2, 2, 2, 1:

22417274880000

3, 2, 2, 1, 1, 1:

1569209241600000

3, 2, 1, 1, 1, 1, 1:

117690693120000000

3, 1, 1, 1, 1, 1, 1, 1:

9415255449600000000

2, 2, 2, 2, 2:

19926466560000

2, 2, 2, 2, 1, 1:

1394852659200000

2, 2, 2, 1, 1, 1, 1:

104613949440000000

2, 2, 1, 1, 1, 1, 1, 1:

8369115955200000000

2, 1, 1, 1, 1, 1, 1, 1, 1:

711374856192000000000

1, 1, 1, 1, 1, 1, 1, 1, 1, 1:

64023737057280000000000

1/2

1, 1:

2, 1:

1, 1, 1:

240

160

3, 1:

1215

2, 2:

960

2, 1, 1:

10920

1, 1, 1, 1:

131040

3125

4, 1:

35840

3, 2:

26460

3, 1, 1:

468720

2, 2, 1:

376320

2, 1, 1, 1:

7136640

1, 1, 1, 1, 1:

142732800

68040

5, 1:

1093750

4, 2:

788480

4, 1, 1:

19676160

3, 3:

714420

3, 2, 1:

14696640

3, 1, 1, 1:

395992800

2, 2, 2:

12096000

2, 2, 1, 1:

321753600

2, 1, 1, 1, 1:

9180864000

1, 1, 1, 1, 1, 1:

275425920000

1647086

6, 1:

35271936

5, 2:

25200000

5, 1, 1:

834750000

4, 3:

21772800

4, 2, 1:

606412800

4, 1, 1, 1:

21714739200

3, 3, 1:

551124000

3, 2, 2:

462672000

3, 2, 1, 1:

16375867200

3, 1, 1, 1, 1:

624558211200

2, 2, 2, 1:

13571712000

2, 2, 1, 1, 1:

512212377600

2, 1, 1, 1, 1, 1:

20534638924800

1, 1, 1, 1, 1, 1, 1:

862454834841600

44040192

7, 1:

1210608210

6, 2:

862202880

6, 1, 1:

36428071680

5, 3:

726468750

5, 2, 1:

26160750000

5, 1, 1, 1:

1192306500000

4, 4:

688128000

4, 3, 1:

22672742400

4, 2, 2:

19301990400

4, 2, 1, 1:

871995801600

4, 1, 1, 1, 1:

42399092736000

3, 3, 2:

17648215200

3, 3, 1, 1:

794573841600

3, 2, 2, 1:

669644236800

3, 2, 1, 1, 1:

32234173171200

3, 1, 1, 1, 1, 1:

1653815773209600

2, 2, 2, 2:

562028544000

2, 2, 2, 1, 1:

26867519078400

2, 2, 1, 1, 1, 1:

1366490655129600

2, 1, 1, 1, 1, 1, 1:

73294276543488000

1, 1, 1, 1, 1, 1, 1, 1:

4104479486435328000

1291401630

8, 1:

44291850240

7, 2:

31525226040

7, 1, 1:

1650182521680

6, 3:

26189412480

6, 2, 1:

1179493539840

6, 1, 1, 1:

66377550919680

5, 4:

24024000000

5, 3, 1:

996360750000

5, 2, 2:

856548000000

5, 2, 1, 1:

47891844000000

5, 1, 1, 1, 1:

2873024154000000

4, 4, 1:

944662118400

4, 3, 2:

747242496000

4, 3, 1, 1:

41621407027200

4, 2, 2, 1:

35513391513600

4, 2, 1, 1, 1:

2113666729574400

4, 1, 1, 1, 1, 1:

134017443495936000

3, 3, 3:

687472077600

3, 3, 2, 1:

32533070169600

3, 3, 1, 1, 1:

1930421593900800

3, 2, 2, 2:

27629291289600

3, 2, 2, 1, 1:

1632369913574400

3, 2, 1, 1, 1, 1:

102593919460032000

3, 1, 1, 1, 1, 1, 1:

6817610009193984000

2, 2, 2, 2, 1:

1375324721971200

2, 2, 2, 1, 1, 1:

85924667372544000

2, 2, 1, 1, 1, 1, 1:

5667816398340096000

2, 1, 1, 1, 1, 1, 1, 1:

392080298676498432000

1, 1, 1, 1, 1, 1, 1, 1, 1:

28229781504707887104000

10:

41250000000

9, 1:

1725958278495

8, 2:

1229098844160

8, 1, 1:

77948119941120

7, 3:

1012612001940

7, 2, 1:

55623543655680

7, 1, 1, 1:

3778500537171360

6, 4:

910486241280

6, 3, 1:

46302881264640

6, 2, 2:

40082087485440

6, 2, 1, 1:

2709710518394880

6, 1, 1, 1, 1:

195827898096230400

5, 5:

879785156250

5, 4, 1:

42522480000000

5, 3, 2:

34054020000000

5, 3, 1, 1:

2294389597500000

5, 2, 2, 1:

1975133160000000

5, 2, 1, 1, 1:

141891750000000000

5, 1, 1, 1, 1, 1:

10836443217600000000

4, 4, 2:

32354677555200

4, 4, 1, 1:

2177210017382400

4, 3, 3:

29927061964800

4, 3, 2, 1:

1726827592089600

4, 3, 1, 1, 1:

123622304053248000

4, 2, 2, 2:

1480403622297600

4, 2, 2, 1, 1:

105692635496448000

4, 2, 1, 1, 1, 1:

8014414222983168000

4, 1, 1, 1, 1, 1, 1:

642437890127953920000

3, 3, 3, 1:

1590810387566400

3, 3, 2, 2:

1361681319398400

3, 3, 2, 1, 1:

96988900200192000

3, 3, 1, 1, 1, 1:

7334436918961152000

3, 2, 2, 2, 1:

82571790292992000

3, 2, 2, 1, 1, 1:

6220219896963072000

3, 2, 1, 1, 1, 1, 1:

494754098732974080000

3, 1, 1, 1, 1, 1, 1, 1:

41352259516058050560000

2, 2, 2, 2, 2:

70105294651392000

2, 2, 2, 2, 1, 1:

5258259760545792000

2, 2, 2, 1, 1, 1, 1:

416068925889576960000

2, 2, 1, 1, 1, 1, 1, 1:

34555232824486133760000

2, 1, 1, 1, 1, 1, 1, 1, 1:

2996316699099530526720000

1, 1, 1, 1, 1, 1, 1, 1, 1, 1:

269668502918957747404800000

1/2

1, 1:

2, 1:

364

1, 1, 1:

2184

5824

3, 1:

45927

2, 2:

34944

2, 1, 1:

413280

1, 1, 1, 1:

4959360

328125

4, 1:

3956736

3, 2:

2748816

3, 1, 1:

52368120

2, 2, 1:

40320000

2, 1, 1, 1:

790020000

1, 1, 1, 1, 1:

15800400000

16901136

5, 1:

287109375

4, 2:

192783360

4, 1, 1:

5276113920

3, 3:

172226250

3, 2, 1:

3710765520

3, 1, 1, 1:

105808459680

2, 2, 2:

2919248640

2, 2, 1, 1:

82356341760

2, 1, 1, 1, 1:

2413385694720

1, 1, 1, 1, 1, 1:

72401570841600

847425747

6, 1:

19205569152

5, 2:

12742537500

5, 1, 1:

467289900000

4, 3:

10745602560

4, 2, 1:

316978421760

4, 1, 1, 1:

12202469959680

3, 3, 1:

283718635200

3, 2, 2:

227980811520

3, 2, 1, 1:

8674473487680

3, 1, 1, 1, 1:

346272172646400

2, 2, 2, 1:

6884179061760

2, 2, 1, 1, 1:

272119771123200

2, 1, 1, 1, 1, 1:

11150350449638400

1, 1, 1, 1, 1, 1, 1:

468314718884812800

42630905856

7, 1:

1239783867861

6, 2:

820472260608

6, 1, 1:

38509515652608

5, 3:

669912271875

5, 2, 1:

25753577850000

5, 1, 1, 1:

1273374402300000

4, 4:

629774745600

4, 3, 1:

21771740390400

4, 2, 2:

17812785070080

4, 2, 1, 1:

871503941468160

4, 1, 1, 1, 1:

44941842712166400

3, 3, 2:

16013289932640

3, 3, 1, 1:

781549566557760

3, 2, 2, 1:

632086200345600

3, 2, 1, 1, 1:

32250094106630400

3, 1, 1, 1, 1, 1:

1718163028905523200

2, 2, 2, 2:

509586171494400

2, 2, 2, 1, 1:

25787106061516800

2, 2, 1, 1, 1, 1:

1361606811018854400

2, 1, 1, 1, 1, 1, 1:

74370257515678924800

1, 1, 1, 1, 1, 1, 1, 1:

4164734420878019788800

2186213819427

8, 1:

79229261709312

7, 2:

52517330521656

7, 1, 1:

3054376965800160

6, 3:

42081147972864

6, 2, 1:

2033695177113600

6, 1, 1, 1:

124734363412538880

5, 4:

38094856800000

5, 3, 1:

1664730767700000

5, 2, 2:

1381949969400000

5, 2, 1, 1:

84008429505000000

5, 1, 1, 1, 1:

5390245096236000000

4, 4, 1:

1565871927459840

4, 3, 2:

1174814652211200

4, 3, 1, 1:

71189531059046400

4, 2, 2, 1:

58513139151667200

4, 2, 1, 1, 1:

3718407517913088000

4, 1, 1, 1, 1, 1:

247617162092947046400

3, 3, 3:

1061731876645440

3, 3, 2, 1:

52697331864777600

3, 3, 1, 1, 1:

3340805087472652800

3, 2, 2, 2:

43137169749273600

3, 2, 2, 1, 1:

2717021464251494400

3, 2, 1, 1, 1, 1:

179177950943902310400

3, 1, 1, 1, 1, 1, 1:

12290784817763150899200

2, 2, 2, 2, 1:

2202952178191564800

2, 2, 2, 1, 1, 1:

144208239116330188800

2, 2, 1, 1, 1, 1, 1:

9812074986152261222400

2, 1, 1, 1, 1, 1, 1, 1:

689218315254477904896000

1, 1, 1, 1, 1, 1, 1, 1, 1:

49623718698322409152512000

10:

115293750000000

9, 1:

5088815388314658

8, 2:

3384762156711936

8, 1, 1:

238110777834209280

7, 3:

2682165579237144

7, 2, 1:

158577126691183200

7, 1, 1, 1:

11760787694595139200

6, 4:

2369380418076672

6, 3, 1:

127364926939810560

6, 2, 2:

106952594279731200

6, 2, 1, 1:

7874091115565875200

6, 1, 1, 1, 1:

611660375537283072000

5, 5:

2278643554687500

5, 4, 1:

115392677400000000

5, 3, 2:

88050925462500000

5, 3, 1, 1:

6461052187590000000

5, 2, 2, 1:

5380989213600000000

5, 2, 1, 1, 1:

414582720552000000000

5, 1, 1, 1, 1, 1:

33512540649588000000000

4, 4, 2:

82938145760870400

4, 4, 1, 1:

6080833510598246400

4, 3, 3:

75358527711667200

4, 3, 2, 1:

4584406440886272000

4, 3, 1, 1, 1:

352124151441334272000

4, 2, 2, 2:

3802122854911180800

4, 2, 2, 1, 1:

290592518588596224000

4, 2, 1, 1, 1, 1:

23282961377232420864000

4, 1, 1, 1, 1, 1, 1:

1945877412699108016128000

3, 3, 3, 1:

4149762953022086400

3, 3, 2, 2:

3437778099429273600

3, 3, 2, 1, 1:

262157195831731200000

3, 3, 1, 1, 1, 1:

20955913889117695488000

3, 2, 2, 2, 1:

215502831872520192000

3, 2, 2, 1, 1, 1:

17126384875877781504000

3, 2, 1, 1, 1, 1, 1:

1418597060949604343808000

3, 1, 1, 1, 1, 1, 1, 1:

121832708781388525240320000

2, 2, 2, 2, 2:

176658089287680000000

2, 2, 2, 2, 1, 1:

13955751928774656000000

2, 2, 2, 1, 1, 1, 1:

1148289199492571136000000

2, 2, 1, 1, 1, 1, 1, 1:

97894300037675581440000000

2, 1, 1, 1, 1, 1, 1, 1, 1:

8599792012711352893440000000

1, 1, 1, 1, 1, 1, 1, 1, 1, 1:

773981281144021760409600000000

1/2

1, 1:

729

2, 1:

3280

1, 1, 1:

19680

209920

3, 1:

1673055

2, 2:

1259520

2, 1, 1:

15057240

1, 1, 1, 1:

180686880

33203125

4, 1:

409108480

3, 2:

277118820

3, 1, 1:

5439234240

2, 2, 1:

4119920640

2, 1, 1, 1:

81761359680

1, 1, 1, 1, 1:

1635227193600

3931876080

5, 1:

68750000000

4, 2:

44490434560

4, 1, 1:

1276531576320

3, 3:

39595897440

3, 2, 1:

872470478880

3, 1, 1, 1:

25558417684800

2, 2, 2:

669865996800

2, 2, 1, 1:

19499050771200

2, 1, 1, 1, 1:

578625151104000

1, 1, 1, 1, 1, 1:

17358754533120000

397725150592

6, 1:

9318941229312

5, 2:

5915674050000

5, 1, 1:

230279349300000

4, 3:

4947877059840

4, 2, 1:

150407060643840

4, 1, 1, 1:

6026073455001600

3, 3, 1:

133971939541440

3, 2, 2:

104567117054400

3, 2, 1, 1:

4154388840201600

3, 1, 1, 1, 1:

169861658808441600

2, 2, 2, 1:

3214336245120000

2, 2, 1, 1, 1:

130544609087692800

2, 1, 1, 1, 1, 1:

5411679488293478400

1, 1, 1, 1, 1, 1, 1:

227290538508326092800

36774046859264

7, 1:

1108410279056080

6, 2:

699060299857920

6, 1, 1:

35105961054003840

5, 3:

563530517550000

5, 2, 1:

22471959510000000

5, 1, 1, 1:

1167947832051000000

4, 4:

528674906439680

4, 3, 1:

18821716270156800

4, 2, 2:

14935226174668800

4, 2, 1, 1:

769313581734297600

4, 1, 1, 1, 1:

41055806672584704000

3, 3, 2:

13332611186294400

3, 3, 1, 1:

685925591176425600

3, 2, 2, 1:

539130653603942400

3, 2, 1, 1, 1:

28534993854945177600

3, 1, 1, 1, 1, 1:

1552877737553747404800

2, 2, 2, 2:

423036228206592000

2, 2, 2, 1, 1:

22234916559266611200

2, 2, 1, 1, 1, 1:

1201885380332833996800

2, 1, 1, 1, 1, 1, 1:

66339150841676224512000

1, 1, 1, 1, 1, 1, 1, 1:

3714992447133868572672000

3230993897342640

8, 1:

121509435167211520

7, 2:

76624549846564720

7, 1, 1:

4790272276607257440

6, 3:

60368573067042240

6, 2, 1:

3042703534313963520

6, 1, 1, 1:

197530855928457891840

5, 4:

54423389020000000

5, 3, 1:

2457069745630500000

5, 2, 2:

1979391966552000000

5, 2, 1, 1:

127400092707888000000

5, 1, 1, 1, 1:

8529806987265564000000

4, 4, 1:

2305640611027353600

4, 3, 2:

1663410184198963200

4, 3, 1, 1:

106862568297195110400

4, 2, 2, 1:

85289295469451673600

4, 2, 1, 1, 1:

5662058291136479232000

4, 1, 1, 1, 1, 1:

388462459976862695424000

3, 3, 3:

1489664977836235200

3, 3, 2, 1:

76228965051650611200

3, 3, 1, 1, 1:

5053806343255673203200

3, 2, 2, 2:

60687111418514380800

3, 2, 2, 1, 1:

3997036134319695667200

3, 2, 1, 1, 1, 1:

272045835327339545472000

3, 1, 1, 1, 1, 1, 1:

19016722917087725846016000

2, 2, 2, 2, 1:

3155496695700494745600

2, 2, 2, 1, 1, 1:

213354808589035137024000

2, 2, 1, 1, 1, 1, 1:

14816101393544690663424000

2, 1, 1, 1, 1, 1, 1, 1:

1050551786590496938395648000

1, 1, 1, 1, 1, 1, 1, 1, 1:

75639728634515779564486656000

10:

276280468750000000

9, 1:

12660245712499967010

8, 2:

8008235242047406080

8, 1, 1:

607005203973600706560

7, 3:

6218462171530528920

7, 2, 1:

385052431637253219840

7, 1, 1, 1:

30363354066354528098880

6, 4:

5458371380680458240

6, 3, 1:

303935724245835601920

6, 2, 2:

248097511528341995520

6, 2, 1, 1:

19411243342603416944640

6, 1, 1, 1, 1:

1583108740851842005401600

5, 5:

5242419799804687500

5, 4, 1:

274113190391040000000

5, 3, 2:

201150249660360000000

5, 3, 1, 1:

15702372002612295000000

5, 2, 2, 1:

12707971231475520000000

5, 2, 1, 1, 1:

1028088432680268960000000

5, 1, 1, 1, 1, 1:

86229336577604220000000000

4, 4, 2:

188876780247239884800

4, 4, 1, 1:

14738565128702813798400

4, 3, 3:

169792220988149760000

4, 3, 2, 1:

10696180102463471616000

4, 3, 1, 1, 1:

863658217151503751577600

4, 2, 2, 2:

8630665696600915968000

4, 2, 2, 1, 1:

692857044374735329689600

4, 2, 1, 1, 1, 1:

57646504929920170328064000

4, 1, 1, 1, 1, 1, 1:

4945887014477709718978560000

3, 3, 3, 1:

9591475771322939894400

3, 3, 2, 2:

7734994202978948505600

3, 3, 2, 1, 1:

620010103358073344716800

3, 3, 1, 1, 1, 1:

51512531739732561738240000

3, 2, 2, 2, 1:

496113215507159298048000

3, 2, 2, 1, 1, 1:

40968756537190603849728000

3, 2, 1, 1, 1, 1, 1:

3488134071728190413168640000

3, 1, 1, 1, 1, 1, 1, 1:

304662054215145799442350080000

2, 2, 2, 2, 2:

396180245601173569536000

2, 2, 2, 2, 1, 1:

32521463327725851967488000

2, 2, 2, 1, 1, 1, 1:

2751732177112880051650560000

2, 2, 1, 1, 1, 1, 1, 1:

238825450478629559871406080000

2, 1, 1, 1, 1, 1, 1, 1, 1:

21157798109060242339984834560000

1, 1, 1, 1, 1, 1, 1, 1, 1, 1:

1904201829815421810598635110400000

1/2

1, 1:

6561

2, 1:

29524

1, 1, 1:

177144

7558144

3, 1:

60407127

2, 2:

45348864

2, 1, 1:

543663120

1, 1, 1, 1:

6523957440

3330078125

4, 1:

41394569216

3, 2:

27762350616

3, 1, 1:

551323773000

2, 2, 1:

415011717120

2, 1, 1, 1:

8276251183200

1, 1, 1, 1, 1:

165525023664000

895132294056

5, 1:

15885009765625

4, 2:

10093234511360

4, 1, 1:

296444174906880

3, 3:

8974375079670

3, 2, 1:

199914163328880

3, 1, 1, 1:

5931368624301120

2, 2, 2:

151615623519360

2, 2, 1, 1:

4483500087306240

2, 1, 1, 1, 1:

133828396956334080

1, 1, 1, 1, 1, 1:

4014851908690022400

180116670346613

6, 1:

4303001657642688

5, 2:

2663032809937500

5, 1, 1:

107261036432550000

4, 3:

2221620956755200

4, 2, 1:

68634437282611200

4, 1, 1, 1:

2810405552904069120

3, 3, 1:

61044986665222560

3, 2, 2:

46813364749831680

3, 2, 1, 1:

1906815455068043520

3, 1, 1, 1, 1:

78956056012513680000

2, 2, 2, 1:

1454409887910819840

2, 2, 1, 1, 1:

59978907826264627200

2, 1, 1, 1, 1, 1:

2501751065687037964800

1, 1, 1, 1, 1, 1, 1:

105073544758855594521600

30187269981732864

7, 1:

930773676708854979

6, 2:

569345004404047872

6, 1, 1:

29835931137157297152

5, 3:

456729180685903125

5, 2, 1:

18600750438870600000

5, 1, 1, 1:

996358599916079400000

4, 4:

428254888124743680

4, 3, 1:

15527711943251988480

4, 2, 2:

12062974391038771200

4, 2, 1, 1:

641954559316772782080

4, 1, 1, 1, 1:

34950741514487621222400

3, 3, 2:

10739313349843478400

3, 3, 1, 1:

571218055027618078080

3, 2, 2, 1:

440753745186916761600

3, 2, 1, 1, 1:

23861451225404394201600

3, 1, 1, 1, 1, 1:

1314327821697648492134400

2, 2, 2, 2:

339828439035307622400

2, 2, 2, 1, 1:

18304604238623986483200

2, 2, 1, 1, 1, 1:

1003702065023219879116800

2, 1, 1, 1, 1, 1, 1:

55737340973875935992217600

1, 1, 1, 1, 1, 1, 1, 1:

3121291094537052415564185600

4485063791172468933

8, 1:

172932110207353356288

7, 2:

105401898543206769504

7, 1, 1:

6918678171068643579600

6, 3:

82421346559071776256

6, 2, 1:

4261495157277350952960

6, 1, 1, 1:

287105204213892412508160

5, 4:

74213913466492800000

5, 3, 1:

3421977639885154350000

5, 2, 2:

2694036086220679200000

5, 2, 1, 1:

180224588286826083000000

5, 1, 1, 1, 1:

12394920846339461430000000

4, 4, 1:

3208909249637693521920

4, 3, 2:

2252796249117173760000

4, 3, 1, 1:

150566166118502688153600

4, 2, 2, 1:

117653229349319894630400

4, 2, 1, 1, 1:

8038939971962526430003200

4, 1, 1, 1, 1, 1:

561699366384552734719180800

3, 3, 3:

2008905345088369247520

3, 3, 2, 1:

104813818609674846672000

3, 3, 1, 1, 1:

7157061370721457247852800

3, 2, 2, 2:

81793815893846658201600

3, 2, 2, 1, 1:

5554369472997809989324800

3, 2, 1, 1, 1, 1:

385844292416374770038860800

3, 1, 1, 1, 1, 1, 1:

27279594615287947713811430400

2, 2, 2, 2, 1:

4306578645725097335193600

2, 2, 2, 1, 1, 1:

297634763018972441231769600

2, 2, 1, 1, 1, 1, 1:

20943229399681895143610572800

2, 1, 1, 1, 1, 1, 1, 1:

1493638606767981905673361920000

1, 1, 1, 1, 1, 1, 1, 1, 1:

107541979687294697208482058240000

10:

614378273437500000000

9, 1:

28906609071577163047407

8, 2:

17635251197159663468544

8, 1, 1:

1409528501788592748625920

7, 3:

13556646859464628956036

7, 2, 1:

864467426719021650164640

7, 1, 1, 1:

71119769638494021340896000

6, 4:

11874017161883294466048

6, 3, 1:

676840085622484896752640

6, 2, 2:

539675294693008078725120

6, 2, 1, 1:

44086823285719712904806400

6, 1, 1, 1, 1:

3714983375894284931813376000

5, 5:

11400335785217285156250

5, 4, 1:

609543214366518432000000

5, 3, 2:

434394934635130969500000

5, 3, 1, 1:

35439562240534752300000000

5, 2, 2, 1:

28044021762242085600000000

5, 2, 1, 1, 1:

2347031627086771825200000000

5, 1, 1, 1, 1, 1:

201646934581996960056000000000

4, 4, 2:

407452944941281522483200

4, 4, 1, 1:

33236473514700396036096000

4, 3, 3:

364173767490523556966400

4, 3, 2, 1:

23474009796305173423718400

4, 3, 1, 1, 1:

1962499902683889618137088000

4, 2, 2, 2:

18541799875224297249177600

4, 2, 2, 1, 1:

1541688663743846288326656000

4, 2, 1, 1, 1, 1:

131605134647154104616615936000

4, 1, 1, 1, 1, 1, 1:

11481290321435710912274890752000

3, 3, 3, 1:

20951078059414439000064000

3, 3, 2, 2:

16545194480968015377715200

3, 3, 2, 1, 1:

1374469973904511509113088000

3, 3, 1, 1, 1, 1:

117242878817959755838129152000

3, 2, 2, 2, 1:

1078161737943490259963904000

3, 2, 2, 1, 1, 1:

91478117087693025678084096000

3, 2, 1, 1, 1, 1, 1:

7933895924162598427734208512000

3, 1, 1, 1, 1, 1, 1, 1:

700344833320879094073016811520000

2, 2, 2, 2, 2:

844696847741867309629440000

2, 2, 2, 2, 1, 1:

71299660204521758068899840000

2, 2, 2, 1, 1, 1, 1:

6152502785810316482180874240000

2, 2, 1, 1, 1, 1, 1, 1:

540470753042130703126875340800000

2, 1, 1, 1, 1, 1, 1, 1, 1:

48142763709758573316902977536000000

1, 1, 1, 1, 1, 1, 1, 1, 1, 1:

4332848733878271598521267978240000000

1/2

1, 1:

59049

2, 1:

265720

1, 1, 1:

1594320

272097280

3, 1:

2176250895

2, 2:

1632583680

2, 1, 1:

19586253960

1, 1, 1, 1:

235035047520

333251953125

4, 1:

4156871147520

3, 2:

2777408868780

3, 1, 1:

55401989020560

2, 2, 1:

41607535656960

2, 1, 1, 1:

831262766181120

1, 1, 1, 1, 1:

16625255323622400

202252053177720

5, 1:

3615783691406250

4, 2:

2277308480778240

4, 1, 1:

67643763250452480

3, 3:

2024409517554060

3, 2, 1:

45334411650702720

3, 1, 1, 1:

1353088310644980000

2, 2, 2:

34177890979814400

2, 2, 1, 1:

1018452243321676800

2, 1, 1, 1, 1:

30482792582808384000

1, 1, 1, 1, 1, 1:

914483777484251520000

80357226117295218

6, 1:

1941310174506812928

5, 2:

1184482734598800000

5, 1, 1:

48628061423773650000

4, 3:

987381471717688320

4, 2, 1:

30774913683524075520

4, 1, 1, 1:

1275106706326797926400

3, 3, 1:

27360151679793834720

3, 2, 2:

20770213504313145600

3, 2, 1, 1:

857945214812787192000

3, 1, 1, 1, 1:

35763602733764022499200

2, 2, 2, 1:

649280755003891046400

2, 2, 1, 1, 1:

27002971249425484185600

2, 1, 1, 1, 1, 1:

1129986441609213873484800

1, 1, 1, 1, 1, 1, 1:

47459430547586982686361600

24211669080726306816

7, 1:

757289091097974563310

6, 2:

454518418318637137920

6, 1, 1:

24455272202861346362880

5, 3:

363965368820461931250

5, 2, 1:

15007869318031112250000

5, 1, 1, 1:

818590005102978517500000

4, 4:

341230859019186339840

4, 3, 1:

12513978870460326604800

4, 2, 2:

9588665940994267545600

4, 2, 1, 1:

520725137493173914828800

4, 1, 1, 1, 1:

28683730095481225949184000

3, 3, 2:

8527931458596204127200

3, 3, 1, 1:

463028034495110492116800

3, 2, 2, 1:

353200619432623844044800

3, 2, 1, 1, 1:

19385595862421566492915200

3, 1, 1, 1, 1, 1:

1075228407014442023664921600

2, 2, 2, 2:

269339629396489666560000

2, 2, 2, 1, 1:

14732969150998045185638400

2, 2, 1, 1, 1, 1:

814844491897300553738649600

2, 1, 1, 1, 1, 1, 1:

45409129366370388701322240000

1, 1, 1, 1, 1, 1, 1, 1:

2542911244516741767274045440000

6032192211375063562530

8, 1:

236516389398061818839040

7, 2:

140911722203274330639240

7, 1, 1:

9554924198118141677605680

6, 3:

109838616987013417601280

6, 2, 1:

5769590279897496724439040

6, 1, 1, 1:

398151458939444862012641280

5, 4:

98867235638890992000000

5, 3, 1:

4622480741149406021250000

5, 2, 2:

3579802674582639960000000

5, 2, 1, 1:

245736079745611923468000000

5, 1, 1, 1, 1:

17189136042066225699462000000

4, 4, 1:

4333863729673323675648000

4, 3, 2:

2987323439779843179724800

4, 3, 1, 1:

204978231938272522944614400

4, 2, 2, 1:

157870847204433477815500800

4, 2, 1, 1, 1:

10991757612676652596386201600

4, 1, 1, 1, 1, 1:

776729724428731110328811520000

3, 3, 3:

2658919452935439016159200

3, 3, 2, 1:

140454333127806639817075200

3, 3, 1, 1, 1:

9776364665128000895827872000

3, 2, 2, 2:

108106869319640856970905600

3, 2, 2, 1, 1:

7493427761773150853950694400

3, 2, 1, 1, 1, 1:

527411115303501647439853632000

3, 1, 1, 1, 1, 1, 1:

37547511711803372432535782400000

2, 2, 2, 2, 1:

5741102666850480887640883200

2, 2, 2, 1, 1, 1:

402595486344048608571856896000

2, 2, 1, 1, 1, 1, 1:

28568984992847254757319598080000

2, 1, 1, 1, 1, 1, 1, 1:

2044818689411157420475292497920000

1, 1, 1, 1, 1, 1, 1, 1, 1:

147226945637603334274221059850240000

10:

1312929706054687500000000

9, 1:

62933351492036176508083545

8, 2:

37424235797295431795343360

8, 1, 1:

3104828650651974018552299520

7, 3:

28630495907760086900190540

7, 2, 1:

1860680539350390155191073280

7, 1, 1, 1:

157594740418671840245813211360

6, 4:

25059507359811501113671680

6, 3, 1:

1451461300419381188189091840

6, 2, 2:

1136563264378706358047293440

6, 2, 1, 1:

95710560550156667639845601280

6, 1, 1, 1, 1:

8243472488561011158927934464000

5, 5:

24057776491355895996093750

5, 4, 1:

1306563249971682022320000000

5, 3, 2:

911770895882955588480000000

5, 3, 1, 1:

76727382846787370550352500000

5, 2, 2, 1:

59716376681678178101640000000

5, 2, 1, 1, 1:

5115994909498122702932880000000

5, 1, 1, 1, 1, 1:

446527777959298486077026400000000

4, 4, 2:

854921766981498438942720000

4, 4, 1, 1:

71939507684434015068566323200

4, 3, 3:

761891224230894268730419200

4, 3, 2, 1:

49861023417628569951643238400

4, 3, 1, 1, 1:

4269430126515969481021898342400

4, 2, 2, 2:

38767584215820600218773094400

4, 2, 2, 1, 1:

3304346327274818678077351526400

4, 2, 1, 1, 1, 1:

287021668055560583742315823104000

4, 1, 1, 1, 1, 1, 1:

25310305007384684105551174041600000

3, 3, 3, 1:

44402926921986835120585867200

3, 3, 2, 2:

34520937216456681605526912000

3, 3, 2, 1, 1:

2941028868295345545501386803200

3, 3, 1, 1, 1, 1:

255369826270900523925908605440000

3, 2, 2, 2, 1:

2274464322727817775081246720000

3, 2, 2, 1, 1, 1:

196656079473474946289406554112000

3, 2, 1, 1, 1, 1, 1:

17268223440944247024854471270400000

3, 1, 1, 1, 1, 1, 1, 1:

1534636964140038468451500427468800000

2, 2, 2, 2, 2:

1757777874327680096629850112000

2, 2, 2, 2, 1, 1:

151358496735683662182132350976000

2, 2, 2, 1, 1, 1, 1:

13239723172368600111371437670400000

2, 2, 1, 1, 1, 1, 1, 1:

1172511825817619231668285238476800000

2, 1, 1, 1, 1, 1, 1, 1, 1:

104813396732971059069479265180057600000

1, 1, 1, 1, 1, 1, 1, 1, 1, 1:

9433205705967395316253133866205184000000

Definition

The simple Hurwitz number $H_{g,\mu}$ counts connected genus-$g$ covers of $\mathbb P^1$ with labelled ramification profile $\mu$ over $\infty$ and simple branching over fixed finite points, each with weight $1/|\operatorname{Aut}(f)|$ [1].

Parameters

$g$

— genus ($g\geq0$)

$\mu$

— ramification profile (a nonempty partition of a positive integer, written as a comma-separated non-increasing list, with $2g-2+\ell(\mu)>0$)

Formulas

(1)

Fix a permutation $\sigma\in S_d$ with cycle type $\mu$. If $N_r(\sigma)$ is the number of ordered $r$-tuples of transpositions $(\tau_1,\ldots,\tau_r)$ with $\tau_1\cdots\tau_r=\sigma$ whose generated subgroup acts transitively on $\{1,\ldots,d\}$, then $\displaystyle H_{g,\mu}=\frac{N_r(\sigma)}{\prod_{i=1}^n\mu_i}$, with $r=2g-2+d+n$ [1].

(2)

For $g=0$ and $n\geq3$, $\displaystyle H_{0,\mu}=(d+n-2)!\,d^{n-3}\prod_{i=1}^n\frac{\mu_i^{\mu_i}}{\mu_i!}$ [1].

(3)

For $g\geq1$ and the one-part partition $(d)$, $\displaystyle H_{g,(d)}=\frac{1}{d\,d!}\sum_{k=0}^{d-1}(-1)^k\binom{d-1}{k}\left(\frac{d(d-1-2k)}{2}\right)^{d+2g-1}$. This is the character expansion of (1) for a $d$-cycle.

Comments

(4)

Each cover is a holomorphic map $f:C\to\mathbb P^1$ from a smooth connected complex curve. Its degree is $d=|\mu|=\sum_i\mu_i$, and $n=\ell(\mu)$ is the number of preimages of $\infty$. There are $r=2g-2+d+n$ fixed distinct finite branch points, each with ramification profile $(2,1,\ldots,1)$, and no other branching. Isomorphisms of covers preserve the map and every label.

(5)

The preimages of $\infty$ are labelled even when ramification indices coincide. Permuting the indices does not change the invariant, so the partition is stored in non-increasing order: $3,1,1$ denotes $(3,1,1)$, and a profile $(1,3,1)$ is found under $3,1,1$. Sorting chooses a representative without dividing the count by a symmetry factor [1].

(6)

Write $m_j$ for the multiplicity of the part $j$ in $\mu$. The convention with unlabelled preimages of $\infty$, used for $h_{g,\mu}$ in [3], is smaller by $|\operatorname{Aut}(\mu)|=\prod_{j\geq1}m_j!$. The finite branch points are fixed, and the entries are not divided by $r!$. The function denoted $H_g(\mu)$ in equation (2.3) of [3] is $H_{g,\mu}/r!$ in this table.

(7)

The inverse-automorphism weight is retained. For $g\geq1$, a double cover with profile $(2)$ has the deck involution and contributes $H_{g,(2)}=1/2$. For profile $(1,1)$ the labels remove that involution from the automorphism group, and $H_{g,(1,1)}=1$. A degree-one cover is an isomorphism, so $H_{g,(1)}=0$ for $g\geq1$.

(8)

The ELSV theorem expresses these counts through intersections of Hodge and cotangent classes on the Deligne-Mumford stack $\overline{\mathcal M}_{g,n}$ with ordered marked points [1]. Its term without a Hodge-class insertion involves intersection numbers of cotangent line classes, and its top Hodge-class term involves $\lambda_g$ Hodge integrals. Stack integration includes automorphism weights. The one-point genus-one integrals $\int_{\overline{\mathcal M}_{1,1}}\psi_1$ and $\int_{\overline{\mathcal M}_{1,1}}\lambda_1$ are both $1/24$, including the factor $1/2$ from the generic elliptic involution.

(9)

The table uses the stable range $2g-2+n>0$. The Hurwitz numbers also exist for $(g,n)=(0,1)$ and $(0,2)$; those cases are used internally by the recurrence but are outside this table.

Programs

(P1)

Python

# Exact one-part simple Hurwitz numbers, with g >= 1 and d >= 1.
from fractions import Fraction
from math import comb, factorial

def one_part_hurwitz(g, d):
    if g < 1 or d < 1:
        raise ValueError("g and d must be positive integers")
    r = d + 2*g - 1
    total = sum((-1)**k * comb(d-1, k)
                * (d*(d-1-2*k)//2)**r for k in range(d))
    return Fraction(total, d*factorial(d))

print(one_part_hurwitz(2, 4))  # 5824

References

[1]

R. Vakil, The moduli space of curves and Gromov-Witten theory, preprint (2006), Sections 3.16–3.19. (arXiv)

[2]

B. Eynard, M. Mulase and B. Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard–Mariño conjecture on Hurwitz numbers, Publications of the Research Institute for Mathematical Sciences 47 (2011), 629–670. (doi)

Links

[3]

Eynard, Mulase and Safnuk: author manuscript with the cut-and-join recurrence and tables of Hurwitz numbers in the Appendix

Similar tables

intersection numbers of cotangent line classes — occur in the ELSV formula as the coefficients with no Hodge-class insertion

$\lambda_g$ Hodge integrals — occur in the ELSV formula as the coefficients with a top Hodge-class insertion

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds every partition of $1\leq d\leq10$ at $0\leq g\leq5$ satisfying $2g-2+\ell(\mu)>0$, giving a complete initial range of low-degree covers in the first six genera)

How they were obtained:

All 793 values are exact rational numbers. The generator evaluates the connected cut-and-join recurrence of [2], Theorem 3.2, in Sage rational arithmetic. Its working quantity is $H_{g,\mu}/r!$, and multiplication by the exact integer $r!$ gives the entries. The unstable degree-one initial value is $H_{0,(1)}=1$.

A separate implementation uses symmetric-group characters computed by the Murnaghan-Nakayama rule, followed by subtraction of disconnected factorizations. It agrees on every entry. Before comparison, its character tables satisfy 3582 orthogonality equations through degree 10. Direct enumeration of permutations and their orbit partitions gives a further check on all 58 stored entries of degrees at most four.

The checks also compare all 103 genus-zero entries with (2), all 138 genus-one entries with the ELSV formula evaluated by the string equation and the $\lambda_1$ multinomial formula, all 50 one-part entries with (3), and 111 examples in [3] after converting the unlabelled convention. Known-answer and deliberately incorrect controls test the comparisons. The same checks are run on values read back from the database; the generator, independent checker and numerical evidence are attached. A further 112 genus-one values agree with ELSV evaluations using frozen API reads of the companion cotangent and Hodge integrals.