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If a permutation has k − 1 descents, then it must be the union of k ascending runs. [44] The number of permutations of n with k ascents is (by definition) the Eulerian number ; this is also the number of permutations of n with k descents.
The permutations of the multiset {,,,, …,,} which have the property that for each k, all the numbers appearing between the two occurrences of k in the permutation are greater than k are counted by the double factorial number ()!!.
The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic combinatorics is used. It also illustrates the parallels in the construction of these two types of numbers, lending support to the binomial-style ...
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. [1] [2] In some cases, cyclic permutations are referred to as cycles; [3] if a cyclic permutation has k elements, it may be called a k-cycle. Some authors widen this definition to include permutations with fixed points in ...
Let K be an algebraic number field with R the ring of integers. The term "Schur's conjecture" refers to the assertion that, if a polynomial f defined over K is a permutation polynomial on R/P for infinitely many prime ideals P, then f is the composition of Dickson polynomials, degree-one polynomials, and polynomials of the form x k.
Every permutation can be produced by a sequence of transpositions (2-element exchanges). Let the following be one such decomposition σ = T 1 T 2... T k. We want to show that the parity of k is equal to the parity of the number of inversions of σ. Every transposition can be written as a product of an odd number of transpositions of adjacent ...
A k-superpattern is a permutation that contains all permutations of length k. For example, 25314 is a 3-superpattern because it contains all 6 permutations of length 3. It is known that k-superpatterns must have length at least k 2 /e 2, where e ≈ 2.71828 is Euler's number, [33] and that there exist k-superpatterns of length ⌈(k 2 + 1)/2 ...
When is a positive integer, () gives the number of n-permutations (sequences of distinct elements) from an x-element set, or equivalently the number of injective functions from a set of size to a set of size .