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Sela Fried

Publications and source records attributed to Sela Fried.

At least 19 recordsLinked to original sources
arXiv2026

Further proofs of conjectures from the OEIS

This is the fourth work in a series devoted to proving conjectures recorded in the On-Line Encyclopedia of Integer Sequences (OEIS). The problems considered here concern elementary and multiplicative number theory, Fibonacci numbers, decimal concatenation, Diophantine and Pell equations, binary representations and bitwise operations, lattice paths, parity patterns, recurrences, and formal power series. Several of the results give complete characterizations of the relevant sequences; others establish exact identities, recurrences, generating functions, asymptotic estimates, integrality properties, or nonoccurrence results. The proofs use combinatorial bijections, congruences, M\"obius inversion, valuations, Fibonacci identities, Pell-type arguments, Lucas' theorem, Riordan arrays, Lagrange inversion, and generating-function methods.

math.GM↗
arXiv2026

Dyck paths on black-and-white lattices

A Dyck path of semilength $n$ is a lattice path from $(0,0)$ to $(n,n)$ consisting of $n$ right-steps $(1,0)$ and $n$ up-steps $(0,1)$ that never rises above the line $y=x$. These paths are enumerated by the Catalan numbers and play a central role in enumerative combinatorics. We color the cells of the integer grid in black and white according to two natural patterns, namely chessboard and column-alternating, and enumerate the Dyck paths having equal numbers of black and white cells beneath them.

math.CO↗
arXiv2026

Proofs of several OEIS conjectures on determinants and permanents

We prove several conjectures recorded in the On-Line Encyclopedia of Integer Sequences. The conjectures considered here concern determinants and permanents of special matrices, such as Toeplitz matrices, cross matrices, Kronecker powers of matrices, and matrices whose entries are defined by powers of differences. As tools we use row and column operations, block determinant formulas, Cauchy determinants, Sylvester's determinant theorem, and $LU$-factorizations. We also obtain closed-form formulas for several related integer sequences for which no such formulas were conjectured.

math.CO↗
arXiv2026

Black Cell Capacity in Catalan polyominoes

A Catalan word is a sequence $w_1w_2\cdots w_n$ of nonnegative integers such that $w_1=0$ and $w_{i}\leq w_{i-1}+1$ for $2\leq i\leq n$. Given a Catalan word, we construct a column-convex polyomino (or \emph{bargraph}) by placing, at position $i$, a column of height $w_i + 1$, with all columns aligned along their bottom edges. On these Catalan polyominoes we define the black cell capacity by coloring the cells in a chessboard pattern and we count the number of black cells in the polyomino. We study the distribution of the black cell capacity over Catalan polyominoes and derive generating functions that encode this statistic.

math.CO↗
arXiv2025

Black-White Cell Capacity in $k$-ary Words and Permutations

We introduce a new bargraph statistic that we call black-white cell capacity. It is obtained by coloring the cells of the bargraph in a chessboard style and recording the numbers of black and white cells contained in the bargraph. We study two word families under this statistic: $k$-ary words and permutations. We obtain the corresponding generating function, in the $k$-ary words case, and a closed-form formula for each $n$, in the permutations case. Of special interest are words containing an equal number of black and white cells, that we call bw-balanced. We obtain generating functions, closed-form formulas, and asymptotics in both cases.

math.CO↗
arXiv2025

Even-up words and their variants

Inspired by OEIS sequence A377912, which consists of the nonnegative integers in which every even digit (except possibly the last) is immediately followed by a strictly larger digit, we define even-up and odd-up words over an alphabet of size~$k$ via similar constraints. We introduce and analyze weak and cyclic variants of these words, deriving explicit generating functions for all eight resulting classes. We then study Catalan words under analogous restrictions. Our results provide new combinatorial interpretations for many integer sequences, including the Motzkin numbers, the Riordan numbers, and the generalized Catalan numbers.

math.CO↗
arXiv2025

Counting k-ary words by number of adjacency differences of a prescribed size

Recently, the general problem of enumerating permutations $\pi=\pi_1\cdots \pi_n$ such that $\pi_{i+r}-\pi_i \neq s$ for all $1\leq i\leq n-r$, where $r$ and $s$ are fixed, was considered by Spahn and Zeilberger. In this paper, we consider an analogous problem on $k$-ary words involving the distribution of the corresponding statistic. Note that for $k$-ary words, it suffices to consider only the $r=1$ case of the aforementioned problem on permutations. Here, we compute for arbitrary $s$ an explicit formula for the ordinary generating function for $n \geq 0$ of the distribution of the statistic on $k$-ary words $\rho=\rho_1\cdots\rho_n$ recording the number of indices $i$ such that $\rho_{i+1}-\rho_i=s$. This result may then be used to find a comparable formula for finite set partitions with a fixed number of blocks, represented sequentially as restricted growth functions. Further, several sequences from the OEIS arise as enumerators of certain classes of $k$-ary words avoiding adjacencies with a prescribed difference. The comparable problem where one tracks indices $i$ such that the absolute difference $|a_{i+1}-a_i|$ is a fixed number is also considered on $k$-ary words and the corresponding generating function may be expressed in terms of Chebyshev polynomials. Finally, combinatorial proofs are found for several related recurrences and formulas for the total number of adjacencies of the form $a(a+s)$ on the various structures.

math.CO↗
arXiv2025

A formula for the number of up-down words

A word $w_1w_2\cdots w_n$ is said to be up-down if $w_1 < w_2 >w_3 \cdots$. Carlitz and Scoville found the generating function for the number of up-down words over an alphabet of size $k$. Using properties of the Chebyshev polynomials we derive a closed-form formula for these numbers.

math.CO↗
arXiv2025

Further results on staircase (cyclic) words

We find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation between all consecutive letters is bounded by $1$. We obtain the analogue results for cyclic words.

math.CO↗
arXiv2024

Further results on staircase graph words

Staircase words are words in which consecutive letters do not differ by more than $1$. We generalize this by extending the restriction to letters lying further apart from each other and obtain the corresponding generating functions, which we express in terms of the Chebyshev polynomials of the second kind.

math.CO↗
arXiv2024

Counting $r\times s$ rectangles in nondecreasing and Smirnov words

The rectangle capacity, a word statistic that was recently introduced by the author and Mansour, counts, for two fixed positive integers $r$ and $s$, the number of occurrences of a rectangle of size $r\times s$ in the bargraph representation of a word. In this work we find the bivariate generating function for the distribution on nondecreasing words of the number of $r\times s$ rectangles and the generating function for their total number over all nondecreasing words. We also obtain the analog results for Smirnov words, which are words that have no consecutive equal letters. This complements our recent results concerned with general words (i.e., not restricted) and Catalan words.

math.CO↗
arXiv2024

Counting $r\times s$ rectangles in (Catalan) words

Generalizing previous results, we introduce and study a new statistic on words, that we call rectangle capacity. For two fixed positive integers $r$ and $s$, this statistic counts the number of occurrences of a rectangle of size $r\times s$ in the bargraph representation of a word. We find the bivariate generating function for the distribution on words of the number of $r\times s$ rectangles and the generating function for their total number over all words. We also obtain the analog results for Catalan words.

math.CO↗
arXiv2023

Staircase graph words

Generalizing the notion of staircase words, introduced by Knopfmacher et.\ al, we define staircase graph words. These are functions $w$ from the vertex set $V$ of a graph into the set $\{1,2,\ldots,k\}$, such that $|w(x)-w(y)|\leq 1$, for every adjacent $x,y\in V$. We find the explicit generating functions for the number of staircase graph words for the grid graph, the rectangle-triangular graph and the king's graph, all of size $2\times n$.

math.CO↗
arXiv2023

Random Walk Labelings of Perfect Trees and Other Graphs

A Random walk labeling of a graph $G$ is any labeling of $G$ that could have been obtained by performing a random walk on $G$. Continuing two recent works, we calculate the number of random walk labelings of perfect trees, combs, and double combs, the torus $C_2\times C_n$, and the graph obtained by connecting three path graphs to form two cycles.

math.CO↗
arXiv2023

On the maximal sum of the entries of a matrix power

Let $p_n$ be the maximal sum of the entries of $A^2$, where $A$ is a square matrix of size $n$, consisting of the numbers $1,2,\ldots,n^2$, each appearing exactly once. We prove that $m_n=\Theta(n^7)$. More precisely, we show that $n(240n^{6}+28n^{5}+364n^{4}+210n^{2}-28n+26-105((-1)^{n}+1))/840\leq p_n\leq n^{3}(n^{2}+1)(7n^{2}+5)/24$.

math.CO↗
arXiv2023

Further Results on Random Walk Labelings

Recently, we initiated the study of random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that each vertex is labeled upon its first visit. In this work, we calculate the number of random walk labelings of several natural graph families: The wheel, fan, barbell, lollipop, tadpole, friendship, and snake graphs. Additionally, we prove several combinatorial identities that emerged during the calculations.

math.CO↗