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Let $S_n$ and $S_{n,k}$ be, respectively, the number of subsets and $k$-subsets of $\mathbb{N}_n=\{1,\ldots,n\}$ such that no two subset elements differ by an element of the set $\mathcal{Q}$. We prove a bijection between such $k$-subsets when $\mathcal{Q}=\{m,2m,\ldots,jm\}$ with $j,m>0$ and permutations $\pi$ of $\mathbb{N}_{n+jm}$ with $k$ excedances satisfying $\pi(i)-i\in\{-m,0,jm\}$ for all $i\in\mathbb{N}_{n+jm}$. We also identify a bijection between another class of restricted permutation and the cases $\mathcal{Q}=\{1,q\}$. This bijection allows us to prove a conjectured recursion relation for the number of such permutations which corresponds to the case $\mathcal{Q}=\{1,4\}$. We also obtain recursion relations for $S_n$ and $S_{n,k}$ in the case $\mathcal{Q}=\{1,5\}$ by first obtaining related recursion relations for the numbers of closed walks of a given length on a particular class of directed pseudograph. We give some classes of $\mathcal{Q}$ for which $S_n$ is also the number of compositions of $n+q$ into a given set of allowed parts, where $q$ is the largest element of $\mathcal{Q}$. A bijection between the $k$-subsets for any $\mathcal{Q}$ and bit strings is also noted. Aided by this, an efficient algorithm for finding $S_n$ and $S_{n,k}$ is given. We also prove a bijection between $k$-subsets for a class of $\mathcal{Q}$ and the set representations of size $k$ of equivalence classes for the occurrence of a given length-($q+1$) subword within bit strings. We then formulate a straightforward procedure for obtaining the generating function for the number of such equivalence classes., Comment: 28 pages, 7 figures. arXiv admin note: text overlap with arXiv:2210.08167

The theories of hypergeometric functions and modular forms are highly intertwined. For example, particular values of truncated hypergeometric functions and hypergeometric character sums are often congruent or equal to Fourier coefficients of modular forms. In this series of papers, we develop and explore an explicit "Hypergeometric-Modularity" method for associating a modular form to a given hypergeometric datum. In particular, for certain length three and four hypergeometric data we give an explicit method for finding a modular form $f$ such that the corresponding hypergeometric Galois representation has a subrepresentation isomorphic to the Deligne representation of $f$. Our method utilizes Ramanujan's theory of elliptic functions to alternative bases, commutative formal group laws, and supercongruences. As a byproduct, we give a collection of eta quotients with multiplicative coefficients constructed from hypergeometric functions. In the second paper, we discuss a number of applications, including explicit connections between hypergeometric values and periods of these explicit eta quotients as well as evaluation formulae for certain special $L$-values., Comment: 65 pages. Several errors in the main results were resolved and the exposition was improved. Version 1 is now divided into two parts, this is part one. The focus of this first paper is to describe the main result clearly with examples. Many applications of our method, including topics from sections 6 and 7 of version 1, will appear in the second part

We consider the restricted subsets of $\mathbb{N}_n=\{1,2,\ldots,n\}$ with $q\geq1$ being the largest member of the set $\mathcal{Q}$ of disallowed differences between subset elements. We obtain new results on various classes of problem involving such combinations lacking specified separations. In particular, we find recursion relations for the number of $k$-subsets for any $\mathcal{Q}$ when $|\mathbb{N}_q-\mathcal{Q}|\leq2$. The results are obtained, in a quick and intuitive manner, as a consequence of a bijection we give between such subsets and the restricted-overlap tilings of an $(n+q)$-board (a linear array of $n+q$ square cells of unit width) with squares ($1\times1$ tiles) and combs. A $(w_1,g_1,w_2,g_2,\ldots,g_{t-1},w_t)$-comb is composed of $t$ sub-tiles known as teeth. The $i$-th tooth in the comb has width $w_i$ and is separated from the $(i+1)$-th tooth by a gap of width $g_i$. Here we only consider combs with $w_i,g_i\in\mathbb{Z}^+$. When performing a restricted-overlap tiling of a board with such combs and squares, the leftmost cell of a tile must be placed in an empty cell whereas the remaining cells in the tile are permitted to overlap other non-leftmost filled cells of tiles already on the board., Comment: 12 pages, 6 figures

We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer $N$) with unit squares and $(1,m-1;t)$-combs for some fixed $m=1,2,\dots$ and $t=2,3,\dots$ that use $n$ tiles in total of which $k$ are combs. A $(1,m-1;t)$-comb is a tile composed of $t$ unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width $m-1$. We show that the entries in the triangle are coefficients of the product of two consecutive generalized Fibonacci polynomials each raised to some nonnegative integer power. We also present a bijection between the tiling of an $(n+(t-1)m)$-board with $k$ $(1,m-1;t)$-combs with the remaining cells filled with squares and the $k$-subsets of $\{1,\ldots,n\}$ such that no two elements of the subset differ by a multiple of $m$ up to $(t-1)m$. We can therefore give a combinatorial proof of how the number of such $k$-subsets is related to the coefficient of a polynomial. We also derive a recursion relation for the number of closed walks from a particular node on a class of directed pseudographs and apply it obtain an identity concerning the $m=2$, $t=5$ instance of the family of triangles. Further identities of the triangles are also established mostly via combinatorial proof., Comment: 25 pages, 8 figures

In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. Using p-adic techniques of Dwork, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long. In particular we prove an analogous version of Long, Tu, Yui, and Zudilin's reduced congruences for each of these six cases. We also conjecture a generalization of Dwork's work which has been observed computationally and which would, together with a proof of modularity for Galois representations associated to our hypergeometric data, yield a full proof of Long's conjectures., Comment: 30 pages, 3 figures

We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by $m-1$ zeros on the right side. The $m=1$ cases are Pascal's triangle and the two families also coincide when $m=2$. Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the $m$-th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the $(1/(1-x^m),x/(1-x))$ Riordan array. Properties of this family of triangles can be obtained quickly as a result. The $(n,k)$-th entry in the $m$-th member of the second family of triangles is the number of tilings of an $(n+k)\times1$ board that use $k$ $(1,m-1)$-fences and $n-k$ unit squares. A $(1,g)$-fence is composed of two unit square sub-tiles separated by a gap of width $g$. We show that the entries in the antidiagonals of these triangles are coefficients of products of powers of two consecutive Fibonacci polynomials and give a bijective proof that these coefficients give the number of $k$-subsets of $\{1,2,\ldots,n-m\}$ such that no two elements of a subset differ by $m$. Other properties of the second family of triangles are also obtained via a combinatorial approach. Finally, we give necessary and sufficient conditions for any Pascal-like triangle (or its row-reversed version) derived from tiling $(n\times1)$-boards to be a Riordan array., Comment: 20 pages, 6 figures

The number of ways to tile an $n$-board (an $n\times1$ rectangular board) with $(\frac12,\frac12;1)$-, $(\frac12,\frac12;2)$-, and $(\frac12,\frac12;3)$-combs is $T_{n+2}^2$ where $T_n$ is the $n$th tribonacci number. A $(\frac12,\frac12;m)$-comb is a tile composed of $m$ sub-tiles of dimensions $\frac12\times1$ (with the shorter sides always horizontal) separated by gaps of dimensions $\frac12\times1$. We use such tilings to obtain quick combinatorial proofs of identities relating the tribonacci numbers squared to one another, to other combinations of tribonacci numbers, and to the Fibonacci, Narayana's cows, and Padovan numbers. Most of these identities appear to be new., Comment: 7 pages, 1 figure