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The\n \n \n $${\\textbf {b}}$$<\/jats:tex-math>\n \n b<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -parking-function polytope\n \n \n $$\\mathfrak {X}_{n}({\\textbf {b}})$$<\/jats:tex-math>\n \n \n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is the convex hull of all\n \n \n $${\\textbf {b}}$$<\/jats:tex-math>\n \n b<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -parking functions of length\n n<\/jats:italic>\n in\n \n \n $$\\mathbb {R}^n$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Geometric properties of\n \n \n $$\\mathfrak {X}_{n}({\\textbf {b}})$$<\/jats:tex-math>\n \n \n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n were previously explored in the specific case where\n \n \n $${\\textbf {b}}=(a,b,b,\\dots ,b)$$<\/jats:tex-math>\n \n \n b<\/mml:mi>\n =<\/mml:mo>\n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and were shown to generalize those of the classical parking-function polytope. In this work, we study\n \n \n $$\\mathfrak {X}_{n}({\\textbf {b}})$$<\/jats:tex-math>\n \n \n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in full generality. We present a minimal inequality and vertex description for\n \n \n $$\\mathfrak {X}_{n}({\\textbf {b}})$$<\/jats:tex-math>\n \n \n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , prove it is a generalized permutahedron, and study its\n h<\/jats:italic>\n -polynomial. Furthermore, we investigate\n \n \n $$\\mathfrak {X}_{n}({\\textbf {b}})$$<\/jats:tex-math>\n \n \n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n through the perspectives of building sets and polymatroids, allowing us to identify its combinatorial types and obtain bounds on its combinatorial and circuit diameters.\n <\/jats:p>","DOI":"10.1007\/s00454-025-00770-1","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T17:20:38Z","timestamp":1757697638000},"page":"339-377","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Combinatorics of Generalized Parking-Function Polytopes"],"prefix":"10.1007","volume":"76","author":[{"given":"Margaret M.","family":"Bayer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steffen","family":"Borgwardt","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Teressa","family":"Chambers","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Spencer","family":"Daugherty","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aleyah","family":"Dawkins","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Danai","family":"Deligeorgaki","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hsin-Chieh","family":"Liao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tyrrell","family":"McAllister","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Angela","family":"Morrison","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Garrett","family":"Nelson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7437-3745","authenticated-orcid":false,"given":"Andr\u00e9s R.","family":"Vindas-Mel\u00e9ndez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,9,12]]},"reference":[{"key":"770_CR1","doi-asserted-by":"crossref","unstructured":"Amanbayeva, A., Wang, D.: The convex hull of parking functions of length $$n$$, Enumer. 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