{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T16:51:58Z","timestamp":1769187118281,"version":"3.49.0"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T00:00:00Z","timestamp":1686614400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T00:00:00Z","timestamp":1686614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2023,8]]},"abstract":"Abstract<\/jats:title>A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (J Comb Theory Ser A 35(2):208\u2013230, 1983) and Kim and Pittel (J Comb Theory Ser A 92(2):197\u2013206, 2000) showed that the number $$S_n$$<\/jats:tex-math>\n \n S<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of score sequences on the complete graph $$K_n$$<\/jats:tex-math>\n \n K<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> satisfies $$S_n=\\Theta (4^n\/n^{5\/2})$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \u0398<\/mml:mi>\n \n (<\/mml:mo>\n \n 4<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n \/<\/mml:mo>\n \n n<\/mml:mi>\n \n 5<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. By combining a recent recurrence relation for $$S_n$$<\/jats:tex-math>\n \n S<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in terms of the Erd\u0151s\u2013Ginzburg\u2013Ziv numbers $$N_n$$<\/jats:tex-math>\n \n N<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with the limit theory for discrete infinitely divisible distributions, we observe that $$n^{5\/2}S_n\/4^n\\rightarrow e^\\lambda \/2\\sqrt{\\pi }$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \n 5<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n S<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \/<\/mml:mo>\n \n 4<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msup>\n \u2192<\/mml:mo>\n \n e<\/mml:mi>\n \u03bb<\/mml:mi>\n <\/mml:msup>\n \/<\/mml:mo>\n 2<\/mml:mn>\n \n \u03c0<\/mml:mi>\n <\/mml:msqrt>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where $$\\lambda =\\sum _{k=1}^\\infty N_k\/k4^k$$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n =<\/mml:mo>\n \n \u2211<\/mml:mo>\n \n k<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \u221e<\/mml:mi>\n <\/mml:msubsup>\n \n N<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \/<\/mml:mo>\n k<\/mml:mi>\n \n 4<\/mml:mn>\n k<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This limit agrees numerically with the asymptotics of $$S_n$$<\/jats:tex-math>\n \n S<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> conjectured by Tak\u00e1cs (J Stat Plan Inference 14(1):123\u2013142, 1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters $$r=2$$<\/jats:tex-math>\n \n r<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$p=e^{-\\lambda }$$<\/jats:tex-math>\n \n p<\/mml:mi>\n =<\/mml:mo>\n \n e<\/mml:mi>\n \n -<\/mml:mo>\n \u03bb<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00493-023-00037-4","type":"journal-article","created":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T08:02:14Z","timestamp":1686643334000},"page":"827-844","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Asymptotic Number of Score Sequences"],"prefix":"10.1007","volume":"43","author":[{"given":"Brett","family":"Kolesnik","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,13]]},"reference":[{"key":"37_CR1","unstructured":"Alekseyev, M.: Proof of Jovovic\u2019s formula, unpublished manuscript available at http:\/\/oeis.org\/A145855\/a145855.txt (2008)"},{"issue":"66","key":"37_CR2","first-page":"18","volume":"21","author":"KS Alexander","year":"2016","unstructured":"Alexander, K.S., Berger, Q.: Local limit theorems and renewal theory with no moments. 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