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38182 bytes, as of the version from 2026-10-16 11:15 (current). Recorded here, not run.
{
"Title": "Simple Hurwitz numbers $H_{g,\\mu}$",
"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 $1/|\\operatorname{Aut}(f)|$ CITE{Vakil}.",
"Parameters": {
"g": {
"type": "Z",
"title": "genus",
"display": "$g$",
"constraints": "$g\\geq0$"
},
"mu": {
"type": "Symbolic",
"title": "ramification profile",
"display": "$\\mu$",
"constraints": "a nonempty partition of a positive integer, written as a comma-separated non-increasing list, with $2g-2+\\ell(\\mu)>0$"
}
},
"Tags": [
"combinatorics",
"algebra"
],
"Links": {
"EMSdata": {
"title": "Eynard, Mulase and Safnuk: author manuscript with the cut-and-join recurrence and tables of Hurwitz numbers in the Appendix",
"url": "https://www.math.ucdavis.edu/static/research/infovault/mulase/RIMS-47-11.pdf"
}
},
"Data properties": {
"type": "Q",
"rigour": "exact",
"complete": "no",
"complete-note": "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",
"rigour details": "All 793 values are exact rational numbers. The generator evaluates the connected cut-and-join recurrence of CITE{EMS}, 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 $H_{0,(1)}=1$.\n\nA 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.\n\nThe checks also compare all 103 genus-zero entries with CITE{genus-zero}, 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 CITE{one-part}, and 111 examples in CITE{EMSdata} 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."
},
"Display properties": {
"number-header": "$H_{g,\\mu}$"
},
"Numbers": {
"0": {
"1,1,1": "24",
"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": {
"1": "0",
"2": "1/2",
"1,1": "1",
"3": "9",
"2,1": "40",
"1,1,1": "240",
"4": "160",
"3,1": "1215",
"2,2": "960",
"2,1,1": "10920",
"1,1,1,1": "131040",
"5": "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",
"6": "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",
"7": "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",
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"1,1,1,1,1,1,1": "862454834841600",
"8": "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",
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"4,2,1,1": "871995801600",
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"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",
"9": "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",
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"3,3,2,1": "32533070169600",
"3,3,1,1,1": "1930421593900800",
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"3,2,1,1,1,1": "102593919460032000",
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"2,2,2,1,1,1": "85924667372544000",
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"1,1,1,1,1,1,1,1,1": "28229781504707887104000",
"10": "41250000000",
"9,1": "1725958278495",
"8,2": "1229098844160",
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"7,3": "1012612001940",
"7,2,1": "55623543655680",
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"3,1,1,1,1,1,1,1": "41352259516058050560000",
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"2,2,2,1,1,1,1": "416068925889576960000",
"2,2,1,1,1,1,1,1": "34555232824486133760000",
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"1,1,1,1,1,1,1,1,1,1": "269668502918957747404800000"
},
"2": {
"1": "0",
"2": "1/2",
"1,1": "1",
"3": "81",
"2,1": "364",
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"4": "5824",
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"5": "328125",
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"6": "16901136",
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"2,2,1,1,1,1,1": "9812074986152261222400",
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"10": "115293750000000",
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"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",
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"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"
},
"3": {
"1": "0",
"2": "1/2",
"1,1": "1",
"3": "729",
"2,1": "3280",
"1,1,1": "19680",
"4": "209920",
"3,1": "1673055",
"2,2": "1259520",
"2,1,1": "15057240",
"1,1,1,1": "180686880",
"5": "33203125",
"4,1": "409108480",
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"2,2,1": "4119920640",
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"1,1,1,1,1": "1635227193600",
"6": "3931876080",
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"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"
}
},
"Comments": {
"parameters": "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.",
"labels": "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 CITE{Vakil}.",
"normalization": "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 CITE{EMSdata}, 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 CITE{EMSdata} is $H_{g,\\mu}/r!$ in this table.",
"automorphisms": "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 $g\\geq1$.",
"intersection-theory": "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 CITE{Vakil}. Its term without a Hodge-class insertion involves HREF{T458}[intersection numbers of cotangent line classes], and its top Hodge-class term involves HREF{T462}[$\\lambda_g$ Hodge integrals]. Stack integration includes automorphism weights; the basic genus-one cotangent and Hodge integrals are $1/24$, including the factor $1/2$ from the elliptic involution.",
"stable-range": "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."
},
"Formulas": {
"monodromy": "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 $r=2g-2+d+n$ CITE{Vakil}.",
"genus-zero": "For $g=0$ and $n\\geq3$, $\\displaystyle H_{0,\\mu}=(d+n-2)!\\,d^{n-3}\\prod_{i=1}^n\\frac{\\mu_i^{\\mu_i}}{\\mu_i!}$ CITE{Vakil}.",
"one-part": "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 CITE{monodromy} for a $d$-cycle."
},
"References": {
"Vakil": {
"bib": "R. Vakil, The moduli space of curves and Gromov-Witten theory, preprint (2006), Sections 3.16–3.19.",
"arxiv": "math/0602347"
},
"EMS": {
"bib": "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": "10.2977/PRIMS/47"
}
},
"Similar tables": [
{
"table": "HREF{T458}[intersection numbers of cotangent line classes]",
"relation": "occur in the ELSV formula as the coefficients with no Hodge-class insertion"
},
{
"table": "HREF{T462}[$\\lambda_g$ Hodge integrals]",
"relation": "occur in the ELSV formula as the coefficients with a top Hodge-class insertion"
}
],
"Programs": {
"one-part-python": {
"language": "Python",
"code": "# Exact one-part simple Hurwitz numbers, with g >= 1 and d >= 1.\nfrom fractions import Fraction\nfrom math import comb, factorial\n\ndef one_part_hurwitz(g, d):\n if g < 1 or d < 1:\n raise ValueError(\"g and d must be positive integers\")\n r = d + 2*g - 1\n total = sum((-1)**k * comb(d-1, k)\n * (d*(d-1-2*k)//2)**r for k in range(d))\n return Fraction(total, d*factorial(d))\n\nprint(one_part_hurwitz(2, 4)) # 5824\n"
}
},
"Keywords": [
"single Hurwitz numbers",
"connected simple Hurwitz numbers",
"transitive transposition factorizations",
"branched covers",
"ELSV formula",
"ramification profiles"
]
}