Arrays with Lexicographic Constraints and Exactly Three Mistakes
Combinatorics OEIS Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is a comprehensive database of integer sequences that has become an indispensable resource for mathematicians, computer scientists, and researchers across various fields. Among its vast collection is sequence A278655, which presents an interesting combinatorial problem related to matrix arrangements with specific ordering constraints.
Key Insight: Sequence A278655 counts the number of n × 7 arrays with entries 0, 1, or 2 that satisfy specific lexicographic ordering conditions while containing exactly three "mistakes" in these ordering constraints.
This sequence belongs to a family of combinatorial problems that explore the interplay between ordering constraints and permissible deviations from those constraints. Understanding such sequences provides insights into error-tolerant systems, coding theory, and the enumeration of constrained matrices.
Sequence A278655 is formally defined as follows7 :
The number of n × 7 0..2 arrays with rows in nondecreasing lexicographic order and columns in nonincreasing lexicographic order, but with exactly three mistakes.
Rows must be in nondecreasing lexicographic order. This means that when comparing each row to the next, it should be lexicographically equal or greater.
Columns must be in nonincreasing lexicographic order. This means that when comparing each column to the next, it should be lexicographically equal or smaller.
In this context, a "mistake" refers to a violation of the specified ordering constraints. The sequence specifically counts arrays that have exactly three such violations. This creates an interesting counting problem where we're enumerating objects that almost satisfy the constraints but deviate in precisely three positions.
The values of sequence A278655 for different values of n (the number of rows) represent the count of 7-column arrays satisfying the described conditions. While the complete sequence data is available in the OEIS7 , we can examine some of its mathematical properties.
| n (rows) | Array Size | Possible Values | Constraints | Mistakes |
|---|---|---|---|---|
| Variable | n × 7 | 0, 1, 2 | Rows: nondecreasing lex order Columns: nonincreasing lex order |
Exactly 3 |
Explore how different array configurations might satisfy or violate the constraints:
Sequences like A278655 have applications in various fields of mathematics and computer science:
Understanding arrays with constrained orderings and permissible errors relates to error-correcting codes and fault-tolerant systems.
This sequence represents a specialized counting problem in enumerative combinatorics with applications to Young tableaux and related structures.
Such sequences help analyze algorithms that work with partially ordered data or that need to handle constrained permutations with errors.
Sequence A278655 is part of a family of sequences that vary parameters such as:
To better understand the structure of arrays counted by A278655, we can visualize how the constraints affect possible configurations.
Sequences like A278655 help mathematicians understand the boundary between strictly ordered structures and those with controlled deviations. This has implications for:
Open questions related to sequences like A278655 include:
OEIS ID: A278655
Description: Number of n×7 0..2 arrays with specific ordering constraints and exactly three mistakes
Field: Combinatorics
Keywords: nondecreasing, nonincreasing, lexicographic, mistakes
The OEIS contains over 300,000 sequences, with new ones added regularly by mathematicians worldwide.
Sequence A278655 is part of a family of sequences exploring the concept of "mistakes" in ordered structures, which has applications in error-correcting codes and fault-tolerant computing.