We present six identities involving Stirling numbers of both kinds. The first two convolution identities appear to be new, and we prove them by algebraic methods using falling powers and generating functions. The other four are recurrence relations for Stirling numbers that recently appeared as conjectures. Two of these were proved, in equivalent forms, elsewhere by algebraic methods. However, we provide novel combinatorial proofs for all of them.
The OEIS sequence A080416, proposed by Barry, is a Stirling-like number triangle defined by a paired decomposition of C(n + 3, 3). The recurrence formula for this sequence can be derived from an article by Corcino et al. However, explicit formulas have not been discovered. In this paper, we derive two formulas for calculating the members of this sequence. One of them involves matrix multiplication, while the other one does not require it. We also generalize the approach for sequences defined by counting North-East lattice paths with weights.
For p, q E N, a finite nonempty set F is said to be (p, q)-Schreier (or maximal (p, q)Schreier, respectively) if q min F >= p F (or q min F = p F , respectively). Using the of (p, q)-Schreier sets of the natural numbers. We show that the counts are taken periodically from Padovan-like sequences that satisfy simple recurrence relations. As an application, we obtain an alternative proof of Beanland et al.'s result. Furthermore, a similar result holds for the counts of maximal (p, q)-Schreier sets. We end with a discussion of the relation between (p, q)-Schreier and maximal (p, q)-Schreier sets.
We find formulas for the determinants of several Toeplitz-Hessenberg matrices whose nonzero entries are Fuss-Catalan numbers. By Trudi's formula, one obtains equivalent multi-sum identities indexed over the set of partitions of a fixed integer involving the product of Fuss-Catalan numbers and multinomial coefficients. We make use of generating functions to establish our results and, as a consequence, several entries from the OEIS are afforded new combinatorial interpretations as determinants of certain Toeplitz-Hessenberg matrices. Finally, we provide counting arguments for several of our determinant identities drawing upon combinatorial interpretations of various specific cases of the Fuss-Catalan numbers in terms of Dyck paths, ternary trees, and non-crossing partitions.
A pseudoprime is a composite integer N that mimics the behavior of primes by satisfying the congruence 2N equivalent to 2 (mod N) in Fermat's little theorem. This paper focuses on the subset of even pseudoprimes and obtains an upper bound for the sum of their reciprocals. Our approach combines analytic arguments with computational verification, showing that this sum is less than 0.0059.
Let s(n) be the number of different remainders n mod k, where 1 <= k <= & LeftFloor;n/2 & RightFloor;. This rather natural sequence is sequence A283190 in the OEIS, and although some basic facts are known, surprisingly, it has been barely studied. First, we prove the asymptotic formula s(n) = cn + O (n/(log n log log n)), where c is an explicit constant. Then we focus on the difference between the consecutive terms s(n) and s(n + 1). It turns out that the value can always increase by at most one, but there exist arbitrarily large decreases. We show that the upper bound on the difference is O(log log n). Finally, we consider "iterated remainder sets". These are related to a problem arising from Pierce expansions, and we prove bounds for the size of these sets as well.
Consider a discrete set of objects and a sample of size N taken with replacement from the set, producing a list of counts of the objects that corresponds to a partition of N. Two statistics that are commonly used for measuring the "diversity" of the sample are the Gini-Simpson index and the Shannon index. We study the number of possible values that these indices can take across all possible partitions of the sample size N as N increases. The two statistics are highly correlated over the set of partitions of N. However, the number of possible values that the Shannon index can take (A383683) far exceeds the number of possible values of the Gini-Simpson index (A069999), with the latter growing quadratically and the former growing faster than every polynomial.
We show in each of the Tribonacci, Padovan, Van der Laan, Perrin, Leonardo, and Narayana's cows sequences that there are infinitely many terms that cannot be represented as the sum of two prime powers.
We study the total positivity of Toeplitz matrices built from generalized hyper-Fibonacci sequences of a fixed generation. Using this approach, we prove that each sequence becomes log-concave beyond a certain point for every generation. We also present several notable special cases that illustrate the scope of our results.
In this work we settle ten conjectures recorded in the On-Line Encyclopedia of Integer Sequences. The results span combinatorics, number theory, and discrete geometry. Our proofs are elementary in spirit and mostly self-contained. We use techniques like combinatorial identities, parity and congruence arguments, generating functions, and elementary geometric and graph-theoretic reasoning. In several cases we also generalize or sharpen the original conjectures.
We show how the software Walnut can be used to obtain concise proofs of results concerning variants of the famous Wythoff game, incorporating blocking maneuvers or additional terminal positions, as discussed by Larsson (2011) and Komak et al. (2025), respectively. Our approach provides automatic proofs that both confirm and extend their results, and the same techniques apply equally to newly introduced variants. Then, using classical techniques, we obtain new recursive and morphic characterizations of Wythoff-type games in which the terminal positions (x, y) satisfy x + y <= & ell;. The use of Walnut in combinatorial game theory is relatively recent, and only a few examples have been investigated so far. The Wythoff game, being directly connected to the Fibonacci numeration system, proves especially well-suited to this kind of approach. It permits us to solve instances for a fixed value of a parameter.
We show that the alternating sum of the floor function of root jn, with j ranging from 1 to n, has an easy evaluation for all odd integers n >= 1. This is in contrast to known non-alternating sums of the same type that hold only for a class of primes. The proof is elementary and was suggested by an AI model. To put this result in perspective, we also prove an asymptotic expression for the analogous sum without the floor function.
The rook numbers are fairly well-studied in the literature. In this paper, we study the max-rook number of the Ferrers boards associated with integer partitions. We show its connections with the Durfee triangle of the partitions. The max-rook number gives a new decomposition of the partition function. We derive the generating functions of the partitions with the Durfee triangle of sizes 3, 4, and 5. We obtain their exact formula and further use it to show the periodicity modulo p for p is an element of N and p >= 2. We also establish their parity and parity bias. We give the growth asymptotics of partitions with the Durfee triangle of sizes 3 and 4. We obtain a new rook analogue of the recurrence relation of the partition function.
Consider a game of permutation wordle in which a player attempts to guess a secret permutation of length n in as few guesses as possible. In each round, the guessing player is told which indices of their guessed permutation are correct. How can we optimize the player's strategy? Samuel Kutin and Lawren Smithline (arXiv:2408.00903) propose a strategy called "cyclic shift" in which all incorrect entries are shifted one index to the right in successive guesses, and they conjecture its optimality. We investigate this conjecture by formalizing what a strategy looks like, performing experimental analysis on inductively constructed strategies, and examining the coefficients of an inductive strategy's generating function.
Parking functions correspond with preferences of n cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. When a car parks in its preferred spot then the corresponding car and corresponding spot are deemed “lucky.” This paper looks briefly at lucky cars which have previously been studied and in simple cases can be understood by a generalization of a result due to Pollak. We also consider lucky spots where the situation is more complex and not previously studied. Probabilities and asymptotics for lucky spots are given for the first few spots on the one-way street. We close with an exploration of the special cases when cars enter the one-way street in either weakly-increasing or weakly-decreasing order of their preferences.
We present four combinatorial proofs of Morgado's formula for the number rho(n) of non-congruent regular integers modulo n, corresponding to sequence A055653 in the On-Line Encyclopedia of Integer Sequences (OEIS), where an integer m is said to be regular modulo n if the congruence m2x equivalent to m (mod n) has a solution x E Z. To illustrate the significance of the sequence and Morgado's formula, we relate them to a recent multi-prime, multi-power generalization of the RSA cryptosystem.
We consider the problem of counting the numbers of functions in various sub-classes of unate and monotone Boolean functions under the restrictions of balancedness and non-degeneracy. Further, we also consider the problem of counting the numbers of inequivalent and NPN-inequivalent functions in these sub-classes.
We study the enumeration of coronas. This counting problem concerns two specific types of lozenge tilings. Exact closed-form formulas for these were conjectured in sequences A380346 and A380416 of the OEIS. We prove these conjectures by employing the weighted adjacency matrix. Furthermore, we extend these results to a general setting.
The arithmetic derivative is a nonlinear derivation on the positive integers which forms a natural analog of the conventional derivative. While exploring solutions to arithmetic differential equations, we stumbled across a curious pattern in the positive integers for which the arithmetic derivative and the Collatz map commute. Here we report on these empirical findings, and prove several analytical results on the form of such numbers. Among these findings is the existence of a family of semiprime numbers which are mapped by the Collatz function to another semiprime having a sum of prime factors which is half of the original semiprime's. We show that this family of semiprimes solves the commutation problem and that the sum of their reciprocals converges.
We investigate positive integer sequences called throwback sequences, generated by moving the initial term of a given sequence to the right a number of places equal to its value, then repeating this step iteratively. Let X be a sequence of distinct positive integers. We prove that each term x of X appears infinitely often in the throwback sequence T (X) of X. Further, we provide an explicit formula for the limiting frequency with which x appears in X. If X is an increasing sequence, we prove that T (X) is uniformly recurrent, i.e., every block of consecutive terms in T(X) appears infinitely often with bounded gaps between consecutive appearances. We discuss how throwback sequences relate to familiar notions such as 2-adic valuations of natural numbers and the Gray code ubiquitous in modern telecommunications. Finally, we examine sorting and mixing properties of the iterated throwback operation in certain special cases.
