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In this paper we introduce and study a new property of infinite words: An infinite word x ∈ A N , with values in a finite set A , is said to be k - self-shuffling ( k ≥ 2 ) if x admits factorizations: x = ∏ i = 0 ∞ U i ( 1 ) ⋯ U i ( k ) = ∏ i = 0 ∞ U i ( 1 ) = ⋯ = ∏ i = 0 ∞ U i ( k ) . In other words, there exists a shuffle of k -copies of x which produces x . We are particularly interested in the case k = 2 , in which case we say x is self-shuffling. This property of infinite words is shown to be independent of the complexity of the word as measured by the number of distinct factors of each length. Examples exist from bounded to full complexity. It is also an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non-self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue–Morse word fixed by the morphism 0 ↦ 01 , 1 ↦ 10 . As another example we show that all Sturmian words of slope α ∈ R ∖ Q and intercept 0 < ρ < 1 are self-shuffling (while those of intercept ρ = 0 are not). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the stepping stone model . One important feature of self-shuffling words stems from their morphic invariance: The morphic image of a self-shuffling word is self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic. [ABSTRACT FROM AUTHOR]

Abstract: In this paper it is shown that there exists a constant such that there do not exist Cameron–Liebler line classes in with parameter x satisfying . This improves all known bounds except for small values of q. [Copyright &y& Elsevier]

Abstract: When a v-set can be equipped with a set of k-subsets so that every 2-subset of the v-set appears in exactly (or at most, or at least) one of the chosen k-subsets, the result is a balanced incomplete block design (or packing, or covering, respectively). For each k, balanced incomplete block designs are known to exist for all sufficiently large values of v that meet certain divisibility conditions. When these conditions are not met, one can ask for the packing with the most blocks and/or the covering with the fewest blocks. Elementary necessary conditions furnish an upper bound on the number of blocks in a packing and a lower bound on the number of blocks in a covering. In this paper it is shown that for all sufficiently large values of v, a packing and a covering on v elements exist whose numbers of blocks differ from the basic bounds by no more than an additive constant depending only on k. [Copyright &y& Elsevier]