Geodesic
On Correlation Polynomials and Subword Complexity
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007).

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We consider words with letters from a $q-ary$ alphabet $\mathcal{A}$. The kth subword complexity of a word $w ∈\mathcal{A}^*$ is the number of distinct subwords of length $k$ that appear as contiguous subwords of $w$. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a precise analysis of the expected $kth$ subword complexity of a randomly-chosen word $w ∈\mathcal{A}^n$. Our other main result describes, for $w ∈\mathcal{A}^*$, the degree to which one understands the set of all subwords of $w$, provided that one knows only the set of all subwords of some particular length $k$. Our methods rely upon a precise characterization of overlaps between words of length $k$. We use three kinds of correlation polynomials of words of length $k$: unweighted correlation polynomials; correlation polynomials associated to a Bernoulli source; and generalized multivariate correlation polynomials. We survey previously-known results about such polynomials, and we also present some new results concerning correlation polynomials. 
@article{DMTCS_2007_special_253_a35,
     author = {Gheorghiciuc, Irina and Ward, Mark Daniel},
     title = {On {Correlation} {Polynomials} and {Subword} {Complexity}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07)},
     year = {2007},
     doi = {10.46298/dmtcs.3553},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3553/}
}
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Gheorghiciuc, Irina; Ward, Mark Daniel. On Correlation Polynomials and Subword Complexity. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007). doi : 10.46298/dmtcs.3553. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3553/

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