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We show that if Xn is a variety of c × n-matrices that is stable under the group Sym ([n]) of column permutations and if forgetting the last column maps Xn into X n−1 , then the number of Sym([n])-orbits on irreducible components of Xn is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FI op -schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one FI op -scheme becomes of product form, where Xn = Y n for some scheme Y in affine c-space. Furthermore, to any FI op -scheme of width one we associate a component functor from the category FI of finite sets with injections to the category PF of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym([n])-orbits of components of Xn, for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem.

We show that if X n {X_{n}} is a variety of c × n {c\times n} -matrices that is stable under the group Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} of column permutations and if forgetting the last column maps X n {X_{n}} into X n - 1 {X_{n-1}} , then the number of Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} -orbits on irreducible components of X n {X_{n}} is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine 𝐅𝐈 𝐨𝐩 {\mathbf{FI^{op}}} -schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one 𝐅𝐈 𝐨𝐩 {\mathbf{FI^{op}}} -scheme becomes of product form, where X n = Y n {X_{n}=Y^{n}} for some scheme Y in affine c-space. Furthermore, to any 𝐅𝐈 𝐨𝐩 {\mathbf{FI^{op}}} -scheme of width one we associate a component functor from the category 𝐅𝐈 {\mathbf{FI}} of finite sets with injections to the category 𝐏𝐅 {\mathbf{PF}} of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} -orbits of components of X n {X_{n}} , for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem. We present applications of our methods to counting fixed-rank matrices with entries in a prescribed set and to counting linear codes over finite fields up to isomorphism.

Aim and objective: The objective of this study was to evaluate patient’s satisfaction about physiotherapy services in urban areas of Pakistan. Materials & Methods: A cross sectional survey study was conducted on 278 patients receiving physiotherapy services at different tertiary care hospitals in urban areas of Pakistan. Simple random sampling technique was used to collect data. The participants were assessed using semi-structured questionnaire. Reliability of questionnaire was also assessed with cronbach’s alpha value (0.9). Data was presented through frequency and percentages of patients responses. Data analysis was done using SPSS. Results: The results of the study showed that out of 278 patients 237 (85.25%) were overall satisfied with the physiotherapy services. Conclusion: This cross sectional survey showed that most of the patients are satisfied with physiotherapy services provided to them at different hospitals in urban areas of Pakistan. The study results concluded that ambiance, duration of treatment session, effectiveness of the treatment and instructions regarding home program were the important factors that affect satisfaction of the patients with physiotherapy services. Keywords: Physiotherapy, satisfaction, physiotherapy services, physiotherapy modalities

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