# Research

### Permuations, Words and Patterns

• Enumeration of words that contain exactly one pattern 123
[pdf] [Maple package Words123New] [Maple package PW123] [Maple package SCHUTZENBERGER]

1) If you want to see the enumeration of words associated with list L and contain exactly one pattern 123, using Theorem 1 in the paper, here are some samples:

the input   yields the output

2) If you want to see the minimal algebraic equations satisfied by the generating functions enumerating words with exactly r occurrences of each of the letters {1, ..., n} that contain exactly one pattern 123, for r=1, r=2, r=3:

the input   yields the output

3) If you want to see the recurrences (in operator notation) for our sequences enumerating words with exactly r occurrences of each of the letters {1, ..., n} that contain exactly one pattern 123, for r=1 and r=2:

the input   yields the output

• Increasing Consecutive Patterns in Words
• ### Partitions

• A Solution to AMM Problem 11908

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