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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A generating function for triangular partitions


Authors: L. Carlitz and Richard Scoville
Journal: Math. Comp. 29 (1975), 67-77
MSC: Primary 10A45
MathSciNet review: 0366803
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {T_k}(n)$ denote the number of solutions in nonnegative integers $ {a_i}$, of the equation

$\displaystyle n = \sum\limits_{i = 1}^k {\sum\limits_{j = 1}^{k - i + 1} {{a_{ij}}} } $

where the $ {a_{ij}}$ satisfy the inequalities $ {a_{ij}} \geqslant {a_{i + 1,j}},{a_{ij}} \geqslant {a_{i + 1,j - 1}}$. We show that

$\displaystyle \sum\limits_{n = 1}^\infty {{T_k}(n){x^n} = {{(1 - x)}^{ - k}}{{(... ...}^{ - k + 1}}{{(1 - {x^5})}^{ - k + 2}} \cdots {{(1 - {x^{2k - 1}})}^{ - 1}}.} $


References [Enhancements On Off] (What's this?)

  • [1] Ronald Alter, Some remarks and results on Catalan numbers, Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1971) Louisiana State Univ., Baton Rouge, La., 1971, pp. 109–132. MR 0329910 (48 #8250)
  • [2] L. Carlitz, Rectangular arrays and plane partitions, Acta Arith. 13 (1967/1968), 29–47. MR 0219430 (36 #2512)
  • [3] L. Carlitz, Sequences, paths, ballot numbers, Fibonacci Quart. 10 (1972), no. 5, 531–549. MR 0317949 (47 #6498)
  • [4] P. A. M. MACMAHON, Combinatory Analysis. Vol. 2, Cambridge, 1916.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0366803-0
PII: S 0025-5718(1975)0366803-0
Keywords: Partitions, generating functions
Article copyright: © Copyright 1975 American Mathematical Society



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