In this paper we determine the number of the meaningful compositions of higher order of the differential operations and Gateaux directional derivative.
Given a two-variable function f without critical points and a compact region R bounded by two level curves of f, this note proves that the integral over R of fs second-order directional derivative in the tangential directions of the interceding level
curves is proportional to the rise in f-value over R. Also discussed are variations on this result when critical points are present or R becomes unbounded.
In this article we consider the cycle structure of compositions of pairs of involutions in the symmetric group S_n chosen uniformly at random. These can be modeled as modified 2-regular graphs, giving rise to exponential generating functions. A compo
sition of two random involutions in S_n typically has about n^(1/2) cycles, and the cycles are characteristically of length n^(1/2). Compositions of two random fixed-point-free involutions, on the other hand, typically have about log n cycles and are closely related to permutations with all cycle lengths even. The number of factorizations of a random permutation into two involutions appears to be asymptotically lognormally distributed, which we prove for a closely related probabilistic model. This study is motivated by the observation that the number of involutions in [n] is (n!)^(1/2) times a subexponential factor; more generally the number of permutations with all cycle lengths in a finite set S is n!^(1-1/m) times a subexponential factor, and the typical number of k-cycles is nearly n^(k/m)/k. Connections to pattern avoidance in involutions are also considered.
Jelinek, Mansour, and Shattuck studied Wilf-equivalence among pairs of patterns of the form ${sigma,tau}$ where $sigma$ is a set partition of size $3$ with at least two blocks. They obtained an upper bound for the number of Wilf-equivalence classes f
or such pairs. We show that their upper bound is the exact number of equivalence classes, thus solving a problem posed by them.
A superdiagonal composition is one in which the $i$-th part or summand is of size greater than or equal to $i$. In this paper, we study the number of palindromic superdiagonal compositions and colored superdiagonal compositions. In particular, we giv
e generating functions and explicit combinatorial formulas involving binomial coefficients and Stirling numbers of the first kind.
A palindromic composition of $n$ is a composition of $n$ which can be read the same way forwards and backwards. In this paper we define an anti-palindromic composition of $n$ to be a composition of $n$ which has no mirror symmetry amongst its parts.
We then give a surprising connection between the number of anti-palindromic compositions of $n$ and the so-called tribonacci sequence, a generalization of the Fibonacci sequence. We conclude by defining a new q-analogue of the Fibonacci sequence, which is related to certain equivalence classes of anti-palindromic compositions
Branko J. Malesevic
,Ivana V. Jovovic
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(2007)
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"The Compositions of the Differential Operations and Gateaux Directional Derivative"
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Branko Malesevic
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