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In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function $\theta_3(q)$ at $q=e^{-\pi}$ and encoded it in an integer sequence $(d(n))_{n\ge0}$ for which he provided a recursive procedure to compute the terms of the sequence. He observed intriguing behaviour of $d(n)$ modulo primes and prime powers. Here we prove (1) that $d(n)$ eventually vanishes modulo any prime power $p^e$ with $p\equiv3$ (mod 4), (2) that $d(n)$ is eventually periodic modulo any prime power $p^e$ with $p\equiv1$ (mod 4), and (3) that $d(n)$ is purely periodic modulo any 2-power $2^e$. Our results also provide more detailed information on period length, respectively from when on the sequence vanishes or becomes periodic. The corresponding bounds may not be optimal though, as computer data suggest. Our approach shows that the above congruence properties hold at a much finer, polynomial level., Comment: 59 pages; AmS-LaTeX; improved results for p congruent to 3 modulo 4; added reference [6]

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroups in finite groups (see Theorems A and C). We also consider the more specialised problems of finding large (non-abelian) nilpotent as well as abelian subgroups in soluble groups.

We show that the generating function $\sum_{n\ge0}M_n\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\sum _{e\ge0} ^{} {z^{4^e}}/( {1-z^{2\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$ modulo $8$ in terms of the binary digits of~$n$, thus improving, respectively complementing earlier results by Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and Yassawi [J. Th\'eorie Nombres Bordeaux 27 (2015), 245-288]. Analogous results are also shown to hold for related combinatorial sequences, namely for the Motzkin prefix numbers, Riordan numbers, central trinomial coefficients, and for the sequence of hex tree numbers., Comment: 28 pages, AmS-LaTeX; minor typos corrected

Generalizing classical extension theory, we solve a Schreier-type extension problem for polygroups by groups. As a consequence, we obtain a method for computing a presentation for a group from its action on a set. The usefulness of this method is illustrated by deriving explicit presentations for the groups $GL_2$ over valuation rings and over valued fields, for the groups $SL_3$ over arbitrary fields, as well as for the five Mathieu groups. Moreover, we sketch some aspects of a new deformation technique for groups, their actions, and presentations, and apply it to compute presentations for the sharply $3$-transitive Zassenhaus groups $M(q^2)$ (in the notation of Huppert and Blackburn) for any odd prime power $q$. This computation serves to demonstrate how suitable deformation of groups and their actions interacts with, and thereby enhances, the presentation method., Comment: 69 pages, AmS-LaTeX

We discuss a partial normalisation of a finite graph of finite groups $(\Gamma(-), X)$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include (i) an important inequality for the number of edges in a Stallings decomposition $\Gamma \cong \pi_1(\Gamma(-), X)$ of a finitely generated virtually free group, (ii) the proof of equivalence of a number of conditions for such a group to be `large', as well as (iii) the classification up to isomorphism of virtually free groups of (free) rank $2$. We also discuss some number-theoretic consequences of the last result., Comment: 16 pages, AmS-LaTeX. arXiv admin note: substantial text overlap with arXiv:1404.1258

In [J. Algebra 452 (2016), 372-389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group $\Gamma$ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining cases, in which the sequence of free subgroup numbers is not ultimately periodic modulo a given prime power, the number of free subgroups of index~$\lambda$ in $\Gamma$ is - essentially - congruent to a binomial coefficient times a rational function in $\lambda$ modulo a power of a prime that divides a certain invariant of the group $\Gamma$, respectively to a binomial sum involving such numbers. These results, apart from their intrinsic interest, in particular allow for a much more efficient computation of congruences for free subgroup numbers in these cases compared to the direct recursive computation of these numbers implied by the generating function results in [J. London Math. Soc. (2) 44 (1991), 75-94]., Comment: AmS-LaTeX; 23 pages

We present a method for obtaining congruences modulo powers of a prime number~$p$ for combinatorial sequences whose generating function satisfies an algebraic differential equation. This method generalises the one by Kauers and the authors [Electron. J. Combin. 8(2) (2012), Art. P37; arXiv:1107.2015] from $p=2$ to arbitrary primes. Our applications include congruences for numbers of non-crossing graphs and numbers of Kreweras walks modulo powers of~$3$, as well as congruences for Fu\ss-Catalan numbers and blossom tree numbers modulo powers of arbitrary primes., Comment: AmS-LaTeX, 35 pages. arXiv admin note: substantial text overlap with arXiv:1107.2015

(Dieudonn\'e and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the coefficients of the exponential of $S(z)$, assuming weaker conditions on the coefficients of $S(z)$ than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the $p$-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbitrary finite Abelian $p$-group, thus generalising numerous results in special cases that had appeared earlier in the literature. Further applications include sufficient conditions for ultimate periodicity of subgroup numbers modulo $p$ for free products of finite Abelian $p$-groups, results on $p$-divisibility of permutation numbers with restrictions on their cycle structure, and a curious "supercongruence" for a certain binomial sum., Comment: AmS-LaTeX; 34 pages; several typos corrected; final version, to appear in Adv. Math

We completely characterise when the sequence of free subgroup numbers of a finitely generated virtually free group is ultimately periodic modulo a given prime power., Comment: AmS-LaTeX; 18 pages

We characterise the modular behaviour of (generalised) Ap\'ery number modulo $9$, thereby in particular establishing two conjectures in "A method for determining the mod-$3^k$ behaviour of recursive sequences" [arXiv:1308.2856]., Comment: AmS-LaTeX, 22 pages

We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers, Eulerian numbers, trinomial coefficients, Delannoy numbers, and to functions counting free subgroups of finite index in the inhomogeneous modular group and its lifts. This leads to numerous new results, including many extensions of known results to higher powers of 3., Comment: AmS-LaTeX, 82 pages

It is shown that the number $f_\lambda$ of free subgroups of index $6\lambda$ in the modular group $\PSL_2(\Z)$, when considered modulo a prime power $p^\al$ with $p\ge5$, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing $\PSL_2(\Z)$ as a special case), and for a one-parameter family of lifts of the Hecke group $\mathfrak{H}(4)=C_2*C_4$. All this is achieved by explicitly determining Pad\'e approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main results complement previous work by Kauers and the authors (arXiv:1107.2015 and ["A method for determining the mod-$3^k$ behaviour of recursive sequences"}, preprint]), where it is shown, among other things, that the free subgroup numbers of $\PSL_2(\Z)$ and its lifts display rather complex behaviour modulo powers of 2 and 3., Comment: AmS-LaTeX; 30 pages; superfluous equation number dropped

We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fu\ss-Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of 2., Comment: AmS-LaTeX; 76 pages; substantial revision: an appendix added which enhances the presented method, many errors and typos corrected; to appear in Electron. J. Combin

We provide exact and asymptotic formulae for the number of unrestricted, respectively indecomposable, $d$-dimensional matrices where the sum of all matrix entries with one coordinate fixed equals 2., Comment: AmS-LaTeX, 9 pages

We prove that the generalised non-crossing partitions associated to well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, respectively by Bessis and Reiner. This manuscript accompanies the paper "Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type" [arXiv:1001.0028], for which it provides the computational details., Comment: AmS-LaTeX, 157 pages; major revision; direct proof that the q-Fusz-Catalan number is a polynomial with non-negative coefficients added

We present the proof of the cyclic sieving conjectures for generalised non-crossing partitions associated to well-generated complex reflection groups due to Armstrong, respectively to Bessis and Reiner, for the 26 exceptional well-generated complex reflection groups. The computational details are provided in the manuscript "Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type - the details" [arXiv:1001.0030]., Comment: AmS-LaTeX, 34 pages; major revision: more efficient presentation; direct proof that the q-Fusz-Catalan numbers are polynomials with non-negative coefficients added

We develop a new setting for the exponential principle in the context of multisort species, where indecomposable objects are generated intrinsically instead of being given in advance. Our approach uses the language of functors and natural transformations (composition operators), and we show that, somewhat surprisingly, a single axiom for the composition already suffices to guarantee validity of the exponential formula. We provide various illustrations of our theory, among which are applications to the enumeration of (semi-)magic squares., Comment: 38 pages, AmS-LaTeX; seriously revised: the relation of this work with prior work of Mat\'i as Menni on an exponential principle in the framework of symmetric monoidal categories is clarified; a section on higher-dimensional magic cubes has been put into the separate paper "A note on higher-dimensional magic matrices"; journal version

Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\Ga_m(q)$ is the free product of two cyclic groups of orders $2m$ respectively $qm$, amalgamated along their subgroups of order $m$. Our main result determines the parity behaviour of the generalized subgroup numbers of $\Ga_m(q)$ which were defined in [T. W. M\"uller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\Ga_m(q)$ into a given finite group $H$, in the case when $\gcd(m,| H|)=1$. This computation depends upon the solution of three counting problems in the Hecke group $\mathfrak H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of $\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$ subgroups of $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_2$ and of $C_\infty$; (iii) determination of the parity of the number of index $n$ subgroups in $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_q$. The first problem has already been solved in [T. W. M\"uller, in: {\it Groups: Topological, Combinatorial and Arithmetic Aspects}, (T. W. M\"uller ed.), LMS Lecture Notes Series 311, Cambridge University Press, Cambridge, 2004, pp. 327-374]. The bulk of our paper deals with the solution of Problems (ii) and (iii)., Comment: AmS-LaTeX; 49 pages; minor corrections, Section 5 restructured for better reading (with some proofs put in a new appendix), new numbering of theorems

We derive an asymptotic expansion for the subgroup of arbitrary Fuchsian groups and some other classes of large groups. Moreover, the main conjecture for Random Walks on symmetric groups is established in full generality. Both problems require subtle estimates of character values and root number multiplicities for symmetric groups, which are also of independent interest., Comment: 57 pages, submitted to Acta Mathematica