Congruences and relations for \(r\)-Fishburn numbers. (English) Zbl 1315.05013
Summary: Recently G. E. Andrews and J. A. Sellers [“Congruences for the Fishburn numbers” (submitted)] proved some amazing congruences for the Fishburn numbers. We extend their results to a more general sequence of numbers. As a result we prove a new congruence mod 23 for the Fishburn numbers and prove their conjectured mod 5 congruence for a related sequence. We also extend and prove some unpublished conjectures of Garthwaite and Rhoades.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 11F20 | Dedekind eta function, Dedekind sums |
| 11P83 | Partitions; congruences and congruential restrictions |
Keywords:
Fishburn numbers; interval orders; congruence; Bernoulli polynomial; Stirling numbers; Glaisher \(T\)-numbersSoftware:
OEISOnline Encyclopedia of Integer Sequences:
Glaisher’s T numbers.Fishburn numbers: number of linearized chord diagrams of degree n; also number of nonisomorphic interval orders on n unlabeled points.
Number of upper triangular zero-one matrices with n ones and no zero rows or columns.
References:
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