An extension of carlitz's bipartition identity

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Abstract

Carlitz' bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function: smax(n1, n2,., nr) = n1 + n2 +. + nr (r - 1)min(n1, n2,., nr). Let π0(n1, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let πl(nl, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which each part has one of the 2r 1 forms: (a + 1, a, a,.a), (a, a + 1, a,., a). (a, a, a,., a+ 1), (ra + 2, ra + 2,., ra + 2), (ra + 3, ra + 3,., ra + 3),., (ra + r, ra + r,., ra + r). Theorem: π0(n1,., nr) = π1(n1,., nr).

Original languageEnglish (US)
Pages (from-to)180-184
Number of pages5
JournalProceedings of the American Mathematical Society
Volume63
Issue number1
DOIs
StatePublished - Mar 1977

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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title = "An extension of carlitz's bipartition identity",
abstract = "Carlitz' bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function: smax(n1, n2,., nr) = n1 + n2 +. + nr (r - 1)min(n1, n2,., nr). Let π0(n1, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let πl(nl, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which each part has one of the 2r 1 forms: (a + 1, a, a,.a), (a, a + 1, a,., a). (a, a, a,., a+ 1), (ra + 2, ra + 2,., ra + 2), (ra + 3, ra + 3,., ra + 3),., (ra + r, ra + r,., ra + r). Theorem: π0(n1,., nr) = π1(n1,., nr).",
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An extension of carlitz's bipartition identity. / Andrews, George E.

In: Proceedings of the American Mathematical Society, Vol. 63, No. 1, 03.1977, p. 180-184.

Research output: Contribution to journalArticle

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T1 - An extension of carlitz's bipartition identity

AU - Andrews, George E.

PY - 1977/3

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N2 - Carlitz' bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function: smax(n1, n2,., nr) = n1 + n2 +. + nr (r - 1)min(n1, n2,., nr). Let π0(n1, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let πl(nl, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which each part has one of the 2r 1 forms: (a + 1, a, a,.a), (a, a + 1, a,., a). (a, a, a,., a+ 1), (ra + 2, ra + 2,., ra + 2), (ra + 3, ra + 3,., ra + 3),., (ra + r, ra + r,., ra + r). Theorem: π0(n1,., nr) = π1(n1,., nr).

AB - Carlitz' bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function: smax(n1, n2,., nr) = n1 + n2 +. + nr (r - 1)min(n1, n2,., nr). Let π0(n1, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let πl(nl, n2,., nr) denote the number of partitions of (n1, n2,., nr) in which each part has one of the 2r 1 forms: (a + 1, a, a,.a), (a, a + 1, a,., a). (a, a, a,., a+ 1), (ra + 2, ra + 2,., ra + 2), (ra + 3, ra + 3,., ra + 3),., (ra + r, ra + r,., ra + r). Theorem: π0(n1,., nr) = π1(n1,., nr).

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