Hecke insertion and maximal increasing and decreasing sequences in fillings of polyominoes. Throughout the first part of the paper we will deal. Intuition tells me that we surely cant find a regular sequence that is increasing and decreasing at the same time. The concepts of decreasing and strictly decreasing sequences.
For example, the content of the moon polyomino at the left of fig. Reasoning indirectly, suppose there were a term an of the sequence such that an l. In either case, fa ngis said to be a monotone sequence. R is a sequence, and if a n fn for n2n, then we write the sequence fas a n or a 1. Now, we cant make another common mistake and assume that because the first few terms increase then whole sequence must also increase.
A sequence an is called strictly increasing if an sequence an is called decreasing if an. We try to establish this result by showing that this sequence is non decreasing. If a sequence is either increasing or decreasing it is called a monotone sequence. We want to show that this sequence is convergent using the. For example, 5 3 1 9 17 23 is a valid v sequence having two elements in the decreasing arm namely 5 and 3, and 3 elements in the increasing arm namely 9, 17 and 23. Math 267 w2018 lecture slides monotone sequences geometric. The length of a partition is the number of its parts, the size of a partition is the sum of its parts. The first 10 terms of this sequence are all increasing and so clearly the sequence cant be a decreasing sequence. Bounds for monotonic sequences each increasing sequence a n is bounded below by a1. Increasing and decreasing sequences proof examsolutions. If a sequence of real numbers is increasing and bounded above, then its supremum is the limit proof. If it is decreasing, then it will converge by the corollary since it is bounded below.
Welcome to the identifying, continuing and describing increasing and decreasing number patterns first 3 numbers shown a math worksheet from the patterning worksheets page at. By the leastupperbound property of real numbers, exists and is finite. Introduction the main purpose of this paper is to put recent results of klazar and noy 10, kasraoui and zeng 9, and chen, wu and yan 2, on the enumeration of 2crossings and 2nestings. For for example, the content of the moon polyomino at the left of fig. Finally, with all this new terminology we can state an important theorem. The following result is an application of the least upper bound property of the real number system. But none of the sequence 6 4 2 or 8 10 15 are v sequence since 6 4 2 has no element in the increasing part while 8 10 15 has no element in the decreasing part. With these specifications an arbitrary composition is split into an alternating sequence of increasing decreasing partitions precisely at the part of the composition in which the midpoint of any of the following patterns occur. Sequences and their limits c frank zorzitto, faculty of mathematics university of waterloo the limit idea for the purposes of calculus, a sequence is simply a list of numbers. Longest increasing subsequence longest increasing subsequence. Typical of the problems we shall treat is the determination of the number of sequences of length n, consisting of the integers 1, 2, m, which have a longest increasing subsequence of length throughout the first part of the paper we will deal only with sequences in which no numbers. If in is a strictly increasing sequence in n then for all n. If it is either strictly increasing or strictly decreasing, we say it is strictly monotone. Qq create a concrete, pictorial, or symbolic increasing decreasing pattern.
Increasing, decreasing and not monotonic sequences. Mat25 lecture 11 notes university of california, davis. Increasing and decreasing number line sequences with missing. A sequence that is either increasing or decreasing is said to be monotone. That is, if extensions vary, they will vary inversely with the intensions.
Unions of increasing and intersections of decreasing. Welcome to the increasing and decreasing number line sequences with missing numbers max. This is a special case of the more general notion of a monotonic function. The following theorem gives a very elegant criterion for a sequence to converge, and explains why monotonicity is so important. A partition is a weakly decreasing sequence of natural numbers, which are called its parts. If a sequence is eventually decreasing, then either 1 the sequence has a lower bound, that is a constant. Example 347 prove that the sequence whose general term is a n pn k0 1 k. We say that a real sequence a n is monotone increasing if n 1 decreasing if n 1 a n 2 monotone non decreasing if n 1 increasing if n 1 a n 2 example. Patterns and relations 7 qq describe an increasing decreasing pattern by stating a pattern rule that includes the starting point and a description of how the pattern continues. Strict peaks 121, 231, 2 weak peak 221 strict valleys 212, 2,312 weak valley 211 as an illustration, we may.
We will prove the theorem for increasing sequences. Let xn be a monotone increasing sequence bounded above and con. Use the buttons below to print, open, or download the pdf version of the identifying, continuing and describing increasing and decreasing number patterns first 3 numbers shown a math worksheet. The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing. Typical of the problems we shall treat is the determination of the number of sequences of length n, consisting of the integers 1, 2, m, which have a longest increasing subsequence of length throughout the first part of the paper we will deal only with sequences in which no numbers are repeated.
The proof is similar to the proof for the case when the sequence is increasing and bounded above, theorem. Increasing and decreasing number line sequences with. So i feel that they are playing with the definition here. A sequence an is called strictly increasing if an sequence an is called decreasing. A sequence is said to be monotone iff it is either monotonically increasing or monotonically decreasing. As examples, the sequence 4 is strictly decreasing, 6 is strictly increasing, while 7 is only increasing since zeros occur in the in. A sequence that is either increasing or decreasing. Increasing and decreasing sequences in fillings of moon. A sequence is bounded if it is bounded from above and below. The sequence 16,8,4,2,1,12, is a decreasing geometric sequence of common ratio a geometric sequence is. This math worksheet was created on 201025 and has been viewed 48 times this week and 65 times this month.
Convergence of a monotone sequence of real numbers lemma 1. With these specifications an arbitrary composition is split into an alternating sequence of increasing decreasing partitions precisely at the part of the composition in which the midpoint of any of the. So we can say that, if terms are arranged in order of in creasing intension, their extensions will be in nonincreasing order. Given a sequence of elements c 1, c 2, c n from a totallyordered universe, find the longest increasing subsequence. A sequence is monotone if it is either increasing or decreasing. In addition, we attempt with partial success to characterize, in intrinsic geometric terms, those convex subsetsc of a convex setx such thatc is not the intersection of any strictly decreasing sequence of convex subsets ofx. Problem 2 7 points determine whether the following sequence is increasing, decreasing or not monotonic. Theorem convergence of monotone, bounded sequences if a sequence is either a increasing and bounded above, or b decreasing and bounded below, then it converges. Once again, since the sequences is bounded from below and decreasing, it is convergent by the monotonic sequence theorem. The content of a moon polyomino is the sequence of column heights, in decreasing order. Typical of the problems we shall treat is the determination of the number of sequences of length n, consisting of the integers 1, 2. Increasing and decreasing sequences in fillings of moon polyominoes 3 definition 2. Hence the middle term which is a constant sequence also converges to 0.
Qq describe an increasing decreasing pattern by stating a pattern rule that includes the starting point and a description of how the pattern continues. Increasing and decreasing sequencesin this tutorial, we show you how to prove whether sequences are increasing or decreasing for all values of n using algebr. Use the buttons below to print, open, or download the pdf version of the increasing and decreasing number line sequences with missing numbers max. It may be printed, downloaded or saved and used in your classroom, home school, or. Prove that upward movement is a continuing pattern, i. If a sequence is eventually increasing, then either 1 the sequence has an upper bound, that is a constant m so that an. As examples, the sequence 4 is strictly decreasing, 6 is strictly increasing.
Recall that a sequence can only be decreasing if all the terms are decreasing. The example we just presented describes an increasing geometric sequence. Each decreasing sequence an is bounded above by a1. It is also bounded between 4 and 7 so by the monotonic sequence theorem, the sequence converges to some number between 4 and 7. Sequences which are either increasing or decreasing are called monotone. This math worksheet was created on 20190109 and has been viewed 198 times this week and 274 times this month. Terms will start to pile up as they keep moving upwardforeward remarks. We prove that a convex setc is a polytope if and only ifc is not the union of any strictly increasing sequence of convex sets. This shows that the least upper bound property that every.
Each decreasing sequence a n is bounded above by a1. Hecke insertion and maximal increasing and decreasing. Here for problems 7 and 8, determine if the sequence is increasing or decreasing by calculating the derivative a0 n. Schensted this paper deals with finite sequences of integers. Subsequences and monotonic sequences department of. It may be printed, downloaded or saved and used in your classroom, home school, or other educational. An increasing sequence that is bounded above will converge to the least upper bound for the.
A sequence of real numbers is also called a real sequence. Pdf compositions of n as alternating sequences of weakly. Determine whether the following sequences are increasing, decreasing or not monotonic. Longest increasing and decreasing subsequences canadian. Identifying, continuing and describing increasing and. Now consider the following sequence s n s 0 3 k s 1 3. This paper deals with finite sequences of integers. Each increasing sequence an is bounded below by a1. A sequence is called monotonic if it is either increasing or decreasing. I was asked on my weekly quiz to find give an example of a sequence that is increasing and decreasing at the same time. The sequence is not monotonic, and is bounded between 2 and 3.
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