Open Problems on Partial Words

Blanchet-Sadri, Francine

doi:10.1007/978-3-540-78291-9_2

Cited by 4 publications

(3 citation statements)

References 107 publications

(162 reference statements)

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“…Note however that Moore et al's tree construction is not suitable for partial words since the border array 001, associated with the canonical string ab , cannot be obtained in such a way. Thus, not only using such a tree structure we cannot get all k = 2 k = 3 k = 4 S h,k (7) = k 6 + k 5 − k 3 if h = 0 88 945 5056 2k 6 + 7k 5 + 5k 4 − 8k 3 + k 2 if h = 1 372 3357 16144 12k 5 Table 3: Formulas for S h,k (7) canonical strings associated with the border arrays, but moreover, some border arrays are missing.…”

Section: Resultsmentioning

confidence: 99%

“…Under these conditions, the previously defined concept of perfect squares can be expressed in terms of the critical positions. Using independent recursive formulas, we compute the exact number of simple and nonsimple critical positions, and in Section 5, we achieve our main goal of calculating the number of bordered partial words of length n with one hole over an alphabet of size k answering an open problem of [5].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Counting Bordered Partial Words by Critical Positions

Allen

Blanchet-Sadri

Byrum

et al. 2011

Electron. J. Combin.

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A partial word, sequence over a finite alphabet that may have some undefined positions or holes, is bordered if one of its proper prefixes is compatible with one of its suffixes. The number theoretical problem of enumerating all bordered full words (the ones without holes) of a fixed length $n$ over an alphabet of a fixed size $k$ is well known. It turns out that all borders of a full word are simple, and so every bordered full word has a unique minimal border no longer than half its length. Counting bordered partial words having $h$ holes with the parameters $k, n$ is made extremely more difficult by the failure of that combinatorial property since there is now the possibility of a minimal border that is nonsimple. Here, we give recursive formulas based on our approach of the so-called simple and nonsimple critical positions.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Counting Bordered Partial Words by Critical Positions

Allen

Blanchet-Sadri

Byrum

et al. 2011

Electron. J. Combin.

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show abstract

“…Partial words were introduced in [BB99], and they too have real-world applications (see [BS08] and references therein). In combinatorics, partial words appear in the context of primitive words [BS05], of (un)avoidability of sets of partial words [BSBK + 09, BBSGR10], and also in the study of the number of squares [HHK08] and overlap-freeness [HHKS09] in (infinite) partial words.…”

Section: Universal Partial Wordsmentioning

confidence: 99%