{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T14:10:25Z","timestamp":1723558225861},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T00:00:00Z","timestamp":1718323200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T00:00:00Z","timestamp":1718323200000},"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":["Period Math Hung"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For a sequence <jats:inline-formula><jats:alternatives><jats:tex-math>$$M=(m_{i})_{i=0}^{\\infty }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msubsup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:mi>i<\/mml:mi>\n                        <\/mml:msub>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mi>i<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mn>0<\/mml:mn>\n                      <\/mml:mrow>\n                      <mml:mi>\u221e<\/mml:mi>\n                    <\/mml:msubsup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of integers such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$m_{0}=1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$m_{i}\\ge 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:tex-math>$$i\\ge 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>i<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, let <jats:inline-formula><jats:alternatives><jats:tex-math>$$p_{M}(n)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>p<\/mml:mi>\n                      <mml:mi>M<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denote the number of partitions of <jats:italic>n<\/jats:italic> into parts of the form <jats:inline-formula><jats:alternatives><jats:tex-math>$$m_{0}m_{1}\\cdots m_{r}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:msub>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>\u22ef<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mi>r<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper we show that for every positive integer <jats:italic>n<\/jats:italic> the following congruence is true: <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} p_{M}(m_{1}m_{2}\\cdots m_{r}n-1)\\equiv 0\\ \\ \\left( \\textrm{mod}\\ \\prod _{t=2}^{r}\\mathcal {M}(m_{t},t-1)\\right) , \\end{aligned}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtable>\n                      <mml:mtr>\n                        <mml:mtd>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>M<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>m<\/mml:mi>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:msub>\n                                <mml:mi>m<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>\u22ef<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>m<\/mml:mi>\n                                <mml:mi>r<\/mml:mi>\n                              <\/mml:msub>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>-<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2261<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mspace\/>\n                            <mml:mspace\/>\n                            <mml:mfenced>\n                              <mml:mtext>mod<\/mml:mtext>\n                              <mml:mspace\/>\n                              <mml:munderover>\n                                <mml:mo>\u220f<\/mml:mo>\n                                <mml:mrow>\n                                  <mml:mi>t<\/mml:mi>\n                                  <mml:mo>=<\/mml:mo>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:mrow>\n                                <mml:mi>r<\/mml:mi>\n                              <\/mml:munderover>\n                              <mml:mi>M<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mo>(<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>m<\/mml:mi>\n                                  <mml:mi>t<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>t<\/mml:mi>\n                                <mml:mo>-<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mfenced>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:mtd>\n                      <\/mml:mtr>\n                    <\/mml:mtable>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:disp-formula>where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {M}(m,r):=\\frac{m}{\\textrm{gcd}\\big (m,\\textrm{lcm}(1,\\ldots ,r)\\big )}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mfrac>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mtext>gcd<\/mml:mtext>\n                        <mml:mrow>\n                          <mml:mo>(<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mi>m<\/mml:mi>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mtext>lcm<\/mml:mtext>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mo>\u2026<\/mml:mo>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mi>r<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                        <mml:mrow>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                      <\/mml:mrow>\n                    <\/mml:mfrac>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Our result answers a conjecture posed by Folsom, Homma, Ryu and Tong, and is a generalisation of the congruence relations for <jats:italic>m<\/jats:italic>-ary partitions found by Andrews, Gupta, and R\u00f8dseth and Sellers.\n<\/jats:p>","DOI":"10.1007\/s10998-024-00579-0","type":"journal-article","created":{"date-parts":[[2024,6,19]],"date-time":"2024-06-19T07:03:42Z","timestamp":1718780622000},"page":"155-167","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["High order congruences for M-ary partitions"],"prefix":"10.1007","volume":"89","author":[{"given":"B\u0142a\u017cej","family":"\u017bmija","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,6,14]]},"reference":[{"key":"579_CR1","volume-title":"The theory of partitions, Reprint of the 1976 original. Cambridge mathematical library","author":"GE Andrews","year":"1998","unstructured":"G.E. Andrews, The theory of partitions, Reprint of the 1976 original. Cambridge mathematical library (Cambridge University Press, Cambridge, 1998)"},{"issue":"4","key":"579_CR2","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s00026-017-0369-6","volume":"21","author":"GE Andrews","year":"2017","unstructured":"G.E. Andrews, E. Brietzke, \u00d8.J. R\u00f8dseth, J.A. Sellers, Arithmetic properties of m-ary partitions without gaps. Ann. Comb. 21(4), 495\u2013506 (2017)","journal-title":"Ann. Comb."},{"key":"579_CR3","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1017\/S0305004100045072","volume":"66","author":"RF Churchhouse","year":"1969","unstructured":"R.F. Churchhouse, Congruence properties of the binary partition function. Proc. Cambridge Philos. Soc. 66, 371\u2013376 (1969)","journal-title":"Proc. Cambridge Philos. Soc."},{"issue":"5","key":"579_CR4","doi-asserted-by":"publisher","first-page":"1482","DOI":"10.1016\/j.disc.2015.12.019","volume":"339","author":"A Folsom","year":"2016","unstructured":"A. Folsom, Y. Homma, J.H. Ryu, B. Tong, On a general class of non-squashing partitions. Discrete Math. 339(5), 1482\u20131506 (2016)","journal-title":"Discrete Math."},{"key":"579_CR5","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1017\/S0305004100050581","volume":"71","author":"H Gupta","year":"1972","unstructured":"H. Gupta, On m-ary partitions. Math. Proc. Camb. Phil. Soc. 71, 343\u2013345 (1972)","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"579_CR6","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.jnt.2016.05.015","volume":"169","author":"QH Hou","year":"2016","unstructured":"Q.H. Hou, H.T. Jin, Y.P. Mu, L. Zhang, Congruences on the number of restricted m-ary partitions. J. Number Theory 169, 79\u201385 (2016)","journal-title":"J. Number Theory"},{"key":"579_CR7","first-page":"939","volume":"30","author":"QL Lu","year":"2010","unstructured":"Q.L. Lu, Z.K. Miao, Congruences for a restricted $$m$$-ary overpartition function. J. Math. Res. Exposit. 30, 939\u2013943 (2010)","journal-title":"J. Math. Res. Exposit."},{"key":"579_CR8","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1017\/S0305004100046259","volume":"68","author":"\u00d8J R\u00f8dseth","year":"1970","unstructured":"\u00d8.J. R\u00f8dseth, Some arithmetical properties of m-ary partitions. Math. Proc. Camb. Phil. Soc. 68, 447\u2013453 (1970)","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"579_CR9","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1006\/jnth.2000.2594","volume":"87","author":"\u00d8J R\u00f8dseth","year":"2001","unstructured":"\u00d8.J. R\u00f8dseth, J.A. Sellers, On m-ary partition function congruences: a fresh look at a past problem. J. Number Theory 87, 270\u2013281 (2001)","journal-title":"J. Number Theory"},{"key":"579_CR10","volume-title":"The mathematica book","author":"S Wolfram","year":"2003","unstructured":"S. Wolfram, The mathematica book, 3rd edn. (Wolfram Media\/Cambridge University Press, Cambridge, 2003)","edition":"3"}],"container-title":["Periodica Mathematica Hungarica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-024-00579-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10998-024-00579-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-024-00579-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,13]],"date-time":"2024-08-13T13:03:12Z","timestamp":1723554192000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10998-024-00579-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,14]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["579"],"URL":"https:\/\/doi.org\/10.1007\/s10998-024-00579-0","relation":{},"ISSN":["0031-5303","1588-2829"],"issn-type":[{"type":"print","value":"0031-5303"},{"type":"electronic","value":"1588-2829"}],"subject":[],"published":{"date-parts":[[2024,6,14]]},"assertion":[{"value":"9 November 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 June 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}