{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T02:06:51Z","timestamp":1775527611242,"version":"3.50.1"},"reference-count":38,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T00:00:00Z","timestamp":1753833600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Exchangeable partitions of the integers and their corresponding mass partitions on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225000282_inline1.png\"\/>\n                        <jats:tex-math>$\\mathcal{P}_{\\infty} = \\{\\mathbf{s} = (s_{1},s_{2},\\ldots)\\colon s_{1} \\geq s_{2} \\geq \\cdots \\geq 0$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225000282_inline2.png\"\/>\n                        <jats:tex-math>$\\sum_{k=1}^{\\infty}s_{k} = 1\\}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    play a vital role in combinatorial stochastic processes and their applications. In this work, we continue our focus on the class of Gibbs partitions of the integers and the corresponding stable Poisson\u2013Kingman-distributed mass partitions generated by the normalized jumps of a stable subordinator with an index\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225000282_inline3.png\"\/>\n                        <jats:tex-math>$\\alpha \\in (0,1)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , subject to further mixing. This remarkable class of infinitely exchangeable random partitions is characterized by probabilities that have Gibbs (product) form. These partitions have practical applications in combinatorial stochastic processes, random tree\/graph growth models, and Bayesian statistics. The most notable class consists of random partitions generated from the two-parameter Poisson\u2013Dirichlet distribution\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225000282_inline4.png\"\/>\n                        <jats:tex-math>$\\mathrm{PD}(\\alpha,\\theta)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . While the utility of Gibbs partitions is recognized, there is limited understanding of the broader class. Here, as a continuation of our work, we address this gap by extending the dual coagulation\/fragmentation results of Pitman (1999), developed for the Poisson\u2013Dirichlet family, to all Gibbs models and their corresponding Poisson\u2013Kingman mass partitions, creating nested families of Gibbs partitions and mass partitions. We focus primarily on fragmentation operations, identifying which classes correspond to these operations and providing significant calculations for the resulting Gibbs partitions. Furthermore, for completion, we provide definitive results for dual coagulation operations using dependent processes. We demonstrate the applicability of our results by establishing new findings for Brownian excursion partitions, Mittag-Leffler, and size-biased generalized gamma models.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.28","type":"journal-article","created":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T08:21:41Z","timestamp":1753863701000},"page":"73-90","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Inverse clustering of Gibbs partitions via independent fragmentation and dual dependent coagulation operators"],"prefix":"10.1017","volume":"63","author":[{"given":"Man-Wai","family":"Ho","sequence":"first","affiliation":[{"name":"The Hong Kong University of Science and Technology"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lancelot F.","family":"James","sequence":"additional","affiliation":[{"name":"The Hong Kong University of Science and Technology"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John W.","family":"Lau","sequence":"additional","affiliation":[{"name":"The University of Western Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,7,30]]},"reference":[{"key":"S0021900225000282_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511617768"},{"key":"S0021900225000282_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-012-0472-x"},{"key":"S0021900225000282_ref17","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v12-434"},{"key":"S0021900225000282_ref20","unstructured":"[20] Ho, M.-W. , James, L. F. and Lau, J. W. (2007). Gibbs partitions (EPPF\u2019s) derived from a stable subordinator are Fox H and Meijer G transforms. Preprint, arXiv:0708.0619 [math.PR]."},{"key":"S0021900225000282_ref5","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v5-63"},{"key":"S0021900225000282_ref1","doi-asserted-by":"publisher","DOI":"10.1214\/17-STS619"},{"key":"S0021900225000282_ref12","doi-asserted-by":"publisher","DOI":"10.1214\/105051606000000655"},{"key":"S0021900225000282_ref21","doi-asserted-by":"publisher","DOI":"10.1017\/jpr.2020.93"},{"key":"S0021900225000282_ref28","first-page":"44","article-title":"A marginal sampler for","volume":"26","author":"Lomeli","year":"2017","journal-title":"J. Comput. Graph. Statist."},{"key":"S0021900225000282_ref34","unstructured":"[34] Pitman, J. (2017). Mixed Poisson and negative binomial models for clustering and species sampling. Unpublished manuscript."},{"key":"S0021900225000282_ref9","first-page":"1","article-title":"The ubiquitous Ewens sampling formula","volume":"31","author":"Crane","year":"2016","journal-title":"Statist. Sci."},{"key":"S0021900225000282_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176342360"},{"key":"S0021900225000282_ref25","doi-asserted-by":"publisher","DOI":"10.3390\/math10040561"},{"key":"S0021900225000282_ref22","doi-asserted-by":"publisher","DOI":"10.1198\/016214501750332758"},{"key":"S0021900225000282_ref2","first-page":"447","article-title":"Ruelle\u015b probability cascades seen as a fragmentation process","volume":"12","author":"Basdevant","year":"2006","journal-title":"Markov Process. Relat. Fields"},{"key":"S0021900225000282_ref27","volume-title":"Special Functions and Their Applications","author":"Lebedev","year":"1972"},{"key":"S0021900225000282_ref37","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1024404422"},{"key":"S0021900225000282_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-43930-2"},{"key":"S0021900225000282_ref24","unstructured":"[24] James, L. F. (2019). Stick-breaking Pitman\u2013Yor processes given the species sampling size. Preprint, arXiv:1908.07186[math.ST]."},{"key":"S0021900225000282_ref14","unstructured":"[14] Gasthaus, J. and Teh, Y. W. (2010). Improvements to the sequence memoizer. In Proc. 24th Int. Conf. Neural Information Processing Systems, Vol. 1, eds. J. Lafferty, C. Williams, J. Shawe-Taylor, R. Zemel, and A. Culotta. ACM, pp. 685\u2013693."},{"key":"S0021900225000282_ref23","doi-asserted-by":"publisher","DOI":"10.1214\/09-AAP660"},{"key":"S0021900225000282_ref11","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2013.217"},{"key":"S0021900225000282_ref36","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-65.2.326"},{"key":"S0021900225000282_ref31","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1022874819"},{"key":"S0021900225000282_ref30","doi-asserted-by":"publisher","DOI":"10.1214\/lnms\/1215453576"},{"key":"S0021900225000282_ref18","doi-asserted-by":"publisher","DOI":"10.1214\/07-AOP377"},{"key":"S0021900225000282_ref7","doi-asserted-by":"publisher","DOI":"10.1111\/rssb.12233"},{"key":"S0021900225000282_ref35","doi-asserted-by":"publisher","DOI":"10.1214\/17-AAP1343"},{"key":"S0021900225000282_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s004400050006"},{"key":"S0021900225000282_ref33","author":"Pitman","year":"2006"},{"key":"S0021900225000282_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s002200050450"},{"key":"S0021900225000282_ref38","doi-asserted-by":"publisher","DOI":"10.1145\/1897816.1897842"},{"key":"S0021900225000282_ref29","volume-title":"The H-Function. Theory and Applications","author":"Mathai","year":"2010"},{"key":"S0021900225000282_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s10958-006-0335-z"},{"key":"S0021900225000282_ref32","first-page":"1","article-title":"Poisson\u2013Kingman partitions","volume":"40","author":"Pitman","year":"2003","journal-title":"Statistics and Science: A Festschrift for Terry Speed"},{"key":"S0021900225000282_ref26","unstructured":"[26] James, L. F , Lee, J. and Ross, N. (2024). Network and interaction models for data with hierarchical granularity via fragmentation and coagulation. Preprint, arXiv:2408.04866 [math.ST]."},{"key":"S0021900225000282_ref19","doi-asserted-by":"publisher","DOI":"10.1214\/08-AOP434"},{"key":"S0021900225000282_ref8","doi-asserted-by":"publisher","DOI":"10.1137\/0133036"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225000282","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:26:03Z","timestamp":1775525163000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225000282\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,30]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225000282"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.28","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,30]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}