The study of pattern avoidance stems from a question in computer science: which sequences of distinct numbers can be ordered by a single pass through a stack? Knuth (1968) found that these sequences are characterized by having no subsequence a, b, c, such that c < a < b. Such a subsequence has the same relative order as the permutation 231, so we say that our original sequence avoids 231.

In more general terms, pattern avoidance is a natural way to restrict the structure of permutations by forbidding subsequences with a certain relative order. This has become a popular topic in enumerative combinatorics, and it has connections to various other fields.

Determining the number of 1324-avoiding permutations of length n is the most important open problem in pattern avoidance. This thesis is comprised of two papers contributing to the inversion monotonicity conjecture by Claesson, Jelínek and Steingrímsson (2012), according to which av_n^k(1324), the number of 1324-avoiders of length n with a fixed number k of inversions, is weakly increasing in n. If the conjecture is true, it improves our understanding of the asymptotic behavior of the number of 1324-avoiders.

In Paper A, we provide an explicit formula for av_n^k(1324) for all n ≥ (k + 7)/2. The proof relies on a novel structural characterization of 1324-avoiders with few inversions. As a byproduct, we show that the inversion monotonicity conjecture holds when n ≥ (k + 7)/2.

In Paper B, we study the inversion monotonicity of classes of permutations avoiding multiple patterns. We show the sets {1324, 231} and {1324, 2314, 3214, 4213} are inversion monotone via explicit injections, and introduce a general procedure for constructing large inversion-monotone sets. We also analyze the limiting structure of large permutations with a fixed number of inversions avoiding 1324 and another pattern of length four, and prove several half-monotonicity results similar to Paper A.

Urbanising landscapes are undergoing rapid transformation as densification, sprawl, land-use competition, and climate change reshape urban green and blue spaces. These dynamics influence not only ecological functions, but also everyday opportunities for people to encounter nature. In particular, children’s access to nearby nature is increasingly recognised as important for wellbeing, environmental awareness, and the development of nature stewardship. However, the conditions that enable children’s meaningful and lasting connections with nature remain unevenly studied. This licentiate thesis examines how formal land use planning and management of urban green-blue infrastructure, and practices of civic organisations, shape children’s opportunities to meaningfully connect with nature in urbanising landscapes. The research is situated within sustainability science and social-ecological systems research and adopts stewardship as an analytical lens for examining how responsibility for human-nature relations is distributed across society. The thesis consists of two qualitative case studies conducted in the Stockholm region. Paper 1 analyses formal land use planning and management processes shaping multifunctional urban green-blue infrastructure, based primarily on semi-structured interviews with municipal and regional officials. Paper 2 investigates how Swedish civic outdoor organisations foster children’s nature connection through pedagogical practices, recurring outdoor activities, and intergenerational learning. The findings show that formal land use planning and management processes play a decisive role in shaping structural conditions for nature access, yet outdoor recreation and children’s everyday contact with nature are often weakly prioritised in the formal governance of land use characterised by sectoral fragmentation and competing land-use interests. At the same time, civic outdoor organisations provide important relational infrastructures that enable children’s repeated, playful, and meaningful engagement with nature. Taken together, the studies demonstrate that children’s opportunities to develop lasting relationships with nature depend on the interaction between structural and relational conditions. Urban land use planning and management determines whether nature is physically available and accessible in a broad sense, while practices of civic organisations contribute to the experiences, knowledge, and meanings through which children learn to relate to landscapes. By linking formal land use planning and management with lived practices of nature engagement, the thesis contributes to sustainability science by highlighting how stewardship emerges across public and civic sectors and by bringing children’s experiences more explicitly into discussions of sustainable urban development.