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Amer. Math. Soc."],"abstract":"<p>\n                    Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schr\u00f6der paths.\n                  <\/p>","DOI":"10.1090\/proc\/16918","type":"journal-article","created":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T14:43:49Z","timestamp":1717685029000},"page":"4141-4154","source":"Crossref","is-referenced-by-count":0,"special_numbering":"784","title":["Billiard partitions, Fibonacci sequences, SIP classes, and quivers"],"prefix":"10.1090","volume":"152","author":[{"given":"Vladimir","family":"Dragovi\u0107","sequence":"first","affiliation":[]},{"given":"Marko","family":"Sto\u0161i\u0107","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,8,26]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/s11139-020-00346-y","article-title":"Combinatorics of the periodic billiards within quadrics","volume":"61","author":"Andrews, George E.","year":"2023","journal-title":"Ramanujan J."},{"key":"2","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1090\/btran\/87","article-title":"Separable integer partition classes","volume":"9","author":"Andrews, George E.","year":"2022","journal-title":"Trans. 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