Increasing and decreasing subsequences and their variants

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Increasing and decreasing subsequences and their variants
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We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2,...,n was obtained by Vershik–Kerov and (almost) by Logan–Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions.
Keywords increasing subsequence decreasing subsequence Young tableau RSK algorithm, Tracy–Widom distribution GUE model pattern avoidance unimodal subsequence alternating subsequence matching oscillating tableau
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most interesting of all The medical College what does this have to do with Increasing and decreasing some sequences Rover Dear emotions that Protection W. corresponds to the spare key to the shape of a purist petition my under the length of the longest increasing subsequent W. is just the longest role length of land that is a way for the 1st draw And the length of the 1 decreasing subsequent is a line in the 1st column that is as example from the previous light 4 1 3 2 wanted this pair the length of the 1st rose to 12 of the 1st row Of the like 1 is increasing subsequent No it's not so easy to actually recovered from these And increasing subsequent of going to De Soto Not such a simple this not just at this 1st rose necessarily such an interesting subject to me this simple example more that answer a The link to the 1st column 3 so the longest decreasing subsidy funds W has like well I want immediate corollary to this by ejection and that's theorem is the 1st year ever approved about 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Permutations W Will get a certain sheds light on the number of Pairs of standard young tableau that they have slammed square and they planned the 1 would be there The length of the 1st right With the length of along increasing subsequent W And so The expectation of the like the ones in prisons Suffolk and just be disarmed that will not as a first-person ask about this question but Torre said about the distribution of ice But in in general and in particular but the rate of broke The expectation that the 1st significant progress was made by Hammersley using sup added ergodic theory he showed 1972 The rate of growth consequences for revenge the limit is Anderson from India You can score would have and exists as some fine I constant which proved was between pi over to him conjectures that true and numerical evidence obtained by longtime collaborators supported this conjecture well this was actually approved Wisconsin is indeed too Independently In 1977 by Logan and chat and by year Girshick Carol exit Logan shot only through the more difficult equality Is greater than or equal to 2 You're Shaquille Ichiro found clever argument using RSK especially the idea that Use this Expression in terms of planned the number of standard young tableau should plan the number of turns in this song is very small Compared to the size of the number of petitions Obama small is is bigger than and factorial is much bigger than the purchase so E. event It don't approximately whenever factor of the largest Turk and being very precise here But this suggest that we want to find a lamb for which have Landed is maximized and figure out what its longest part the largest part you can take this maxim discrete maximization problem here And by using the someplace When an goes infinity you get a continuous problem Texas Variations but we I want to do is find this curve worry normalized area the curved ones so that we minimize logarithm of integrating Over under the curve At every point Distant West Mr. Ozal but the Problem reduced to and this was what Solved by looking chapter there should care or not so easy to guess this Curtis Turned back Pretty good eyesight to see that this is the equation given parametric played by But these equations as White to co-sign data And wants to reply signed of mind status signed retreat at 0 upon the fact that it should be exit taxis to Merely proves this constant Is greater than or equal to 2 and I mentioned that The year should Caro Found this very nice Proof using to very elementary argument last year because of us solve the problem of the expectation and the next step is to try to get even better information but the actual distribution of the length of the longest increasing subsequent what to do that you need some kind of formula Distribution in case and let you caravan number Permutations such that The longest increasing such secrets of W like west put out And here because I'm going to be looking for relating total Trinity what was does it mean to be clear so we need some kind of Expression for this In order to Do it analytically tribes see what happens The committee now the 1st such result again do that Hammersley 72 It was at tour van The number of permutations with no Increasing substitute can't of 1 3 is travel and so this is 1 of the many Car motorial interpretations of cattle in the 1st minute another life Collecting these interpretations and But she now I have over 140 of them and there's way each rougher Higher values It's not so simple but Ira Gasol was the 1st of 4 over 1998 a generating function for these numbers fixed can select the look of this generating function some of the UK access to an Turns out In fact square The nominative the result intended to be just Take a bite determinant of church kind of that doesn't look so promising 1st the complicated actually extract good Asymptotic information from so 1 tables True death Dubai to determine the source of a determined is equal to expand that out And that's easy to see That indeed this is a generating function for Thailand numbers recovering the result of merciless now in arresting result the corollary to Gesell formula is that although there's no simple formula for UK event in general for richer each day
You can be computed Quickly It's so what's known as a P recursive functions fix It satisfies linear Returned with Colin no middle coefficients and years but you for a new fiber plant from the likes of course aren't you figure out what occurrences are The jury quick Column time to compute the value of 3rd there some interesting conjectures about the exact form of injured curses due to bearish of for her Take so Now that we have a Sort The These numbers UK event we can try to understand references and Infiniti and this is the pioneering work of Dies Council day from the limiting distribution properly scaled While the length of the longest increase subsequent misses also discussed just Perceived by Mrs. The flying of the finest Distribution hinted so-called of a true functions Look at this non-linear differential equations tailor-made to equations and He wanted the journey class The pressure equations that were classified by Tampa Bay and with certain Meshoppen dishes I'm Such its asymptotic say what happens if you this standard solution if you get rid of the nonlinear termed Harry function so I will need this Function to define this Tracy rhythm distribution before I do that on presents a little bit about The mathematician 4 I because he had a very interesting wife who was born in Paris in 1863 and who was an excellent mathematicians from shown by winning the concrete PC holds the key to 1890 in 1908 she was the 1st person in history of passenger in an airplane and set White duration record 17 minutes prick 2 years Silas Willie held a position of equivalent of being prime minister for him and to and the story he died 1933 we are going back to the mathematics will defined in terms of this panel of a function of X The probability distributions Tracy would on distribution FFT to is this book is completely over the air few have seen for why would anybody Think of such a probability distribution But it turned out this is exactly video distribution that describes Limiting behavior of a length of longest increase in subsequent would introduce scalar properly subtract being witches as prices were would've and abide by the standard deviation which turns out of order and the 1 Sex and the probability that this quantity Roster Any real number In goes infinity You have to proceed with distribution so this was a spectacular breakthrough in this subject and various nice corollary can be drawn about these distribution of increasing such sequences in particular No The length of along with that actually should be easy This to be the expectations of the E Expected length along the increasing subsequent remember whose asymptotic Twice as were revenge and this result Issued 2nd term constant times and the 1 6 iMac constant is the point in terms of the treaty with distribution about minus point what minus 1 . 7 7 1 So tiny you prove this Huron well Just over did all the comic torso North Florida I'm concerned after just analysis were Seth 99 per cent of the proof and blocked the details there but we could ask says Tracy when distribution which be written here would elevate function here But added Forward defined when big Director Hansen proved the so where did this come from 1st place this is going to give another 1 of these unexpected connections of increasing subsequent other topics as random matrices So Road person died kindness yesterday they dousing unitary ensemble a certain probability distribution on permission matrices and by an matrices New ways to find this way Not so circuit in here the exact Definition the system Marie standard natural Probability distributions matrices and Tracy rhythm showed that the largest I get value of random permission matrix chosen from the GUE model Properly stale normalized this way Also satisfies Tracy with distribution so here we have seemed like 2 completely different ways in which this division has arisen And in terms of increasing subsequent as their distribution and I can values the largest item value permission matrix we asked this connection just a coincidence Wurzer some connection Nanda The connection is was actually given by 100 Congo after 4 days of fuels metal here The what was about what he did Namely how you look at the fury of rant on apologies surfaces and the reason that are be true interpretations 2 places where the Tracy with distribution appears has to do with the fact that there are 2 ways and surface can be described as became glow Polygons along the the edges and build up the surface in that way and there This way In this theory of quantum gravity that said about this way the lamp surfaces shown be connected random matrices we can also think of a surface as a ramify covering others fear and then they come torch Permutations comes in 2 The description of the surface so extremely brief indication of why these 2 subjects related but days That finishes
Highlights The subject of increasing and decreasing subsequent permutations that we can think about generalizations and analyze finance many of these are discussed in my paper for this conference now I just . 1 years which is a joint work with these 4 Chinese co-authors Bill Chen even was Messina do and Katherine this is wrong will replace permutations With another commenter object matter so complete matching get to end points and will join them by hand arts no to courts with a common vertex here is complete matching she wants a way of providing joined points Walks To be teach sure them how to 0 properties matches that of matters that would be interested in a really pairs of edges of the match and crossings like this to adjust cross their interiors The last thing on 1 edge This contained MCI's now the total number of matching on the rocket to and which which means the numbers from 1 of the 2 It's easy to see that it too and minus 1 semi factorial the product of all the odd numbers from 1 to 2 and have another interpretation cavalier numbers as a number of matches on these 2 endpoints with no crossing and also number of matching with no nesting surprisingly both times Now I will give a sketch of proof of this theorem body Doing by Jackson between these matches if you wanna count those with no crossing and those with no mess things in something that's well known to be countered by Calin numbers namely so-called ballot sequences sequences of 2 and 1 thousand minus where every parcel Summers non-native total song is 0 so and wants an end minus 1 3rd them water so that all parcels on While the by Jackson's between Imagine no crossings in these about sequences very simple are Matching without any crossings we just label every left-handed and point guard by 1 every right in and by minors Very easy to and this gives a by ejection the sequence is about sequence give the by junction between these matching about is kind of surprising in exactly the same rule works for Matches with next of what points labeled by 1 of the right hand and point by minus 1 and still is a by ejection ballots answers to think about that for a few minutes at least 2 were that's true but the saliva debt Account numbers matter what would like to do it will develop a theory for matching analogous increasing and decreasing subsequent Permutations so why should be the correct analog well A 1st attempt would be that we can't think of a matching the very obviously has a fixed point your emotions actually it's a kind of permutations which is labeled a vertices 1 and 2 0 Numbers of being matched They were posed by a fixed point free pollution so just get this permutation W. Young associated with our matching Would shares And cycles of to ever did They Lewis justifying the increasing and decreasing subsequent of a matching the beating of the course funding increasing and decreasing subsequent as of this permutation well The water in this definition is no longer in symmetry between the 2 Statistics for permutations of obvious that increasing subsequent decreasing subsequent as have the same distribution contributors reversed permutations But actually restrict the permutations The bill Fixed-point free evolution these distributions are actually different So we knew something else and this is what turned out for the most number of mutually crossing edges In the most number of mutually NASCAR So Loser Match N C ground across a number of them will be the largest share So that there's a cake frosting so that there exists Kelly edges any jewel which cancer-related nesting number Of the match in any event will be the largest Such that into this picture but actually turns out that the last number is 1 half of the length of the longest decreasing subsequent this Permutations here Show The Associated imagines actually decreasing subsequent now is a good 1 But we need start increasing subsequently replaced crossed low It's certainly not clear that these are the same distribution like permutations were increasing and decreasing subsequent obviously and distribution money that this is the case of Not only do they have the same distribution but they have a symmetric Joint Distribution Aprilette half of an applied The number of matching 2 endpoints crossing number I and Nestor number jet At end S & J so particular A the number of matches with a during crossing number because a number of nesting aII the number matches the giving nests in check so the case tables z wrote When there were no crossings in domestic life Did previously shown that they were both equal to the cattle and number in here we have this generalization but Unlike the case for permutations were increasing and decreasing subsequent as obviously of the distribution and don't know a simple reason this is true arresting problem on the simple by Judge and matching transforms crossing number into the nest number What we have to do here is go through Are K Type argument pointed him on options that So the main tool now was going to be a generalization of standard young Temple which Oracle oscillating we can't think of a standard John tableau as a way of building up A diagram of a partition 1 square at a time The deal be square in scene and below which the 1 those his 1st than we had another square where the 2 of them in another square with 3 gone before oscillating tableau
You can add or remove which players and staff so here we have an arson tabloid we always or start empty shell each year in 8 steps and ended up at the shade 3 1 This shed 3 1 9 8 Every step by either lawyer Taken away always maintaining a dynamic partitions but There's no really simple would represent the 1 Kind of tableaux like for standard don't need this whole cuts will now How do oscillating below commented the theory of matches There is our matching an example of a matching red do something on almost completely analogous to what I did The RSK algorithm which we encoded permutation by a special kind of matching his matching his right hand and Contra war and we label the right hand and points of these Park From 1 day and from right to left 1 2 3 right hand and . decreasing order from left to right of left and points of the same label has the right hand and points and there going do the same procedure that we did for scary Wigan scandalous sequence of numbers from left to right the return receipt number for the 1st time that left The aim of the park He inserted by the RSK insertion will I'm received for the 2nd time we deleted search for work Answer to delete 4 Insert 3 sir want 3 lead to We sure going to get a certain sequence Health care We only want to remember their shaky that will be an oscillating tableau each step We knew at once were removed 1 square Is not Slater blow 2 steps in empty shell starts out with him to ship ends up with which I will be the oscillating Babylon Correspondent to this match and it's not hard to see bit tricky because Only a memory not the numbers in side of these shapes here but it turns out just from this knowledge of his shaky recoverable of these numbers in the entire match This man from matching oscillating By Jackson from all the matches on and point to be passing several pointed to an attempt to show wrote doughnuts a quick word about representation theory comes into this picture same way that it came into permutations picture the number of oscillating tableau of linked to an empty shell of them would be the number of matching on 1 of the 2 and is to and minus 1 double factorial That is the dimension of Brouwer algebra centralized algebra refined or some sector groups Operating on tend to power defining representation and this is a wrong algebra representation theory that plays a role rustling fell below the symmetric group Plays for standard tabloid but I won't need thinking more about that made sure we need here Ashens Hero from matching for permutations We apply desire algorithm However read off the length of the longest increasing and decreasing sequences and a completely analogous dirt There was true for matching we have matching him We look at the oscillating tableau them that we get The crossing number of em would be this is just saying the length of the longest Call any of these shade don't know where the necessary she wonders Pierce adding and removing Scrooge at every step of river does appear that will be crossing number Nesting number will be the length of the longest all these ships and the idea of the proof is eternal saying about really except that it would we can reduce it to what Mary RISK algorithm in new the ordinary Xinjiang fell for instance go back to this For example here The length of longest columnist 2 batters sea crossing number indeed too is the most number of Edges that pairwise crossed Incirlik delight along this road wished to that's going to be nest number and indeed these to mass edges and no more row now dividend or given this emotions said Is it a citywide across As number are average symmetric Joint Distribution supposed Elmhurst Crossing number nesting numbers Jed applying this out this Isolating Cabello algorithm here Now we can define a new national to be the 1 that corresponds to the answer will get a conjugate every shit still isolating tableau obviously transposed every should we still are just having removing square steps Why meet this a by the book By that Fuser by ejection will resist you need to prime such a few them trying to this castle blog but clearly If we transpose shape change the length of this column like The longest Rob So why should says theorem crossing number from of Jed be the new length of wanders column hearing last time so this map Prime is involution matching in urging the across nested was kind of complicated Andalusian and don't know us Some politeness Without going through this process Caballo ejection would be interesting to find a more direct injection And we now approved 0 that 2 distributions these 2 statistics have metric Joint Distribution And as open find analog of simple permutation reversals in a changing IS India what is an analogous Ejection interchanges the mast cross But chaser Now we can start thinking about the Christian the enumerating Matching with given nesting crossing number this is the ideologue over gasoline did for this You may have and given this determined Basel functions once we have such an enumeration we can try the figure of the asymptotic Behaviors in the Infiniti so but say that him as a matching crossing number and which can now means that in this
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