{"id":"https://openalex.org/W7211879296","doi":"https://doi.org/10.48550/arxiv.2609.04979","title":"A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks","display_name":"A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks","publication_year":2026,"publication_date":"2026-09-04","ids":{"openalex":"https://openalex.org/W7211879296","doi":"https://doi.org/10.48550/arxiv.2609.04979"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.04979","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04979","pdf_url":"https://arxiv.org/pdf/2609.04979","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.04979","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5150637924","display_name":"Hao Yu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Yu, Hao","raw_affiliation_strings":["Department of Mathematics , National University of Singapore , Singapore 119076 , Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , National University of Singapore , Singapore 119076 , Singapore","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5136957376","display_name":"Louxin Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Zhang, Louxin","raw_affiliation_strings":["Department of Mathematics , National University of Singapore , Singapore 119076 , Singapore"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , National University of Singapore , Singapore 119076 , Singapore","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I165932596"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.5396000146865845,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.5396000146865845,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10015","display_name":"Genomics and Phylogenetic Studies","score":0.3257000148296356,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10385","display_name":"Plant Diversity and Evolution","score":0.015399999916553497,"subfield":{"id":"https://openalex.org/subfields/1105","display_name":"Ecology, Evolution, Behavior and Systematics"},"field":{"id":"https://openalex.org/fields/11","display_name":"Agricultural and Biological Sciences"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/tree-child-networks","display_name":"tree-child networks","score":0.9769999980926514},{"id":"https://openalex.org/keywords/phylogenetic-networks","display_name":"phylogenetic networks","score":0.8069999814033508},{"id":"https://openalex.org/keywords/reticulation","display_name":"reticulation","score":0.7599999904632568},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"combinatorial proof","score":0.8600000143051147},{"id":"https://openalex.org/keywords/recurrence-relations","display_name":"recurrence relations","score":0.4410000145435333}],"concepts":[{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.711899995803833},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.6941999793052673},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.6736000180244446},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6029999852180481},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5803999900817871},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4814000129699707},{"id":"https://openalex.org/C193252679","wikidata":"https://www.wikidata.org/wiki/Q242125","display_name":"Phylogenetic tree","level":3,"score":0.4715999960899353},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.4668000042438507},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4350000023841858},{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.4146000146865845},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.3840999901294708},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3763999938964844},{"id":"https://openalex.org/C2987825933","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorial analysis","level":2,"score":0.30640000104904175},{"id":"https://openalex.org/C53553401","wikidata":"https://www.wikidata.org/wiki/Q47307","display_name":"Genealogy","level":1,"score":0.29249998927116394},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.2870999872684479},{"id":"https://openalex.org/C2776020190","wikidata":"https://www.wikidata.org/wiki/Q997453","display_name":"New class","level":2,"score":0.27889999747276306},{"id":"https://openalex.org/C26619641","wikidata":"https://www.wikidata.org/wiki/Q3142246","display_name":"Phylogenetic network","level":4,"score":0.2777000069618225},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2533999979496002}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.04979","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04979","pdf_url":"https://arxiv.org/pdf/2609.04979","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.04979","doi":"https://doi.org/10.48550/arxiv.2609.04979","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.04979","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.04979","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.04979","pdf_url":"https://arxiv.org/pdf/2609.04979","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322724","display_name":"Ministry of Education, India","ror":"https://ror.org/048xjjh50"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7211879296.pdf","grobid_xml":"https://content.openalex.org/works/W7211879296.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W7124243998","https://openalex.org/W7162333096","https://openalex.org/W7204143617","https://openalex.org/W7223774734"],"related_works":["https://openalex.org/W7160968794","https://openalex.org/W4235374327","https://openalex.org/W3156174818","https://openalex.org/W7204143617","https://openalex.org/W7165788579","https://openalex.org/W3213746956","https://openalex.org/W4414991658","https://openalex.org/W2061017441","https://openalex.org/W3134831432","https://openalex.org/W3034821689"],"abstract_inverted_index":{"Tree-child":[0],"networks":[1,26],"are":[2],"a":[3,43],"useful":[4],"class":[5],"of":[6,24,47],"binary":[7],"phylogenetic":[8],"networks.":[9],"The":[10],"Pons--Batle":[11],"identity":[12],"(Pons":[13],"and":[14],"Batle,":[15],"\\textit{Scientific":[16],"Reports},":[17],"2021)":[18],"states":[19],"that":[20],"the":[21],"number":[22],"$a_{n,k}$":[23],"tree-child":[25],"with":[27],"$k$":[28],"reticulations":[29],"on":[30],"$n$":[31],"taxa":[32],"satisfies":[33],"\\[":[34],"a_{n,k}=(n-k+1)a_{n,k-1}":[35],"+\\frac{n(2n+k-3)}{n-k}a_{n-1,k}.":[36],"\\]":[37],"In":[38],"this":[39,48],"paper,":[40],"we":[41],"present":[42],"short":[44],"combinatorial":[45],"proof":[46],"identity.":[49]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-08T00:00:00"}