Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features. We also use third-party cookies that help us analyze and understand how you use this website. The cookie is used to store the user consent for the cookies in the category "Other. First, the n represents the times we can use parentheses. So form the recursive function using the above two cases. For example, lets say we want to map a list of User entities to a UserDTO list. Enter the choices on each line in the generator and click on generate button. I know that there might not be an extremely fast way to do this, but I'd like to at least be able to run it with some reasonably small numbers and have it finish in a relatively short time period and not take up more RAM than it has too. Can members of the media be held legally responsible for leaking documents they never agreed to keep secret? Why hasn't the Attorney General investigated Justice Thomas? I've rewritten the previous solution fully in Java and more user friendly. What kind of tool do I need to change my bottom bracket? So, we have a set {1,2,3,4,5,6} and we want three elements. Algorithm to get the combinations of all items in array JavaScript; All combinations of sums for array in JavaScript; Generate all combinations of supplied words in JavaScript; Generating combinations from n arrays with m elements in JavaScript; Python program to get all pairwise combinations from a list I appreciate it, but can I know why you are using Lists.newLinkedList instead of List copy = new LinkedList<>(); is this version anymore efficient. Feel free to revert. Example: All clothes combinations among 3 colors and 5 sizes: 1st multiple choice, the color red, green, blue 2nd multiple choice, the size XS S M L XL: The total number is equal to the multiplication of the number of each possible choices. Let's say all your lists are in lists, which is a list of lists. Are we not on stackoverflow after all? Can be replaced with a List. See also: Cartesian product of an arbitrary number of sets. You need recursion: Let's say all your lists are in lists, which is a list of lists. I used a Set simply to get rid of any duplicates. It can easily be altered for lists. What happens when XML parser encounters an error? The map has two values (a key and value), while a List only has one value (an element). the order is not important (ab = ba) empty set is not considered. Which is the best library to generate combinations in Java? Why hasn't the Attorney General investigated Justice Thomas? How to generate the power-set of a given List? Then click on 'download' to download all combinations as a txt file. For example, given the following lists: X: [A, B, C] Y: [W, X, Y, Z] Then I should be able to generate 12 combinations: [AW, AX, AY, AZ, BW, BX, BY, BZ, CW, CX, CY, CZ] If a third list of 3 elements were added, I'd have 36 combinations, and so forth. Click Kutools > Insert > List All Combinations, see screenshot: 2. can one turn left and right at a red light with dual lane turns? For example, given the following lists: One of my problems is that my array myght vary from 3 to 15 different arrays and each myght be empty (I might add a 0 just not to be empty) or have many values. Then all the specified values and separators have been listed into the dialog box, see screenshot: 4 .And then click Ok button, and a prompt box will pop out to remind you select a cell to . Iteratively, in the same way, calculate the permutations until. NB: Generation limited to 10000 combinations. Home > Validation > Dependent > Tables. Analytical cookies are used to understand how visitors interact with the website. Create a recursive function that accepts a string (s), count of opening brackets (o) and count of closing brackets (c) and the value of n. if the value of opening bracket and closing bracket is equal to n then print the string and return. These generate the next combination from the previous and avoid repetitions. How to print and connect to printer using flutter desktop via usb? Let result be the list of your required permutations. Why is a "TeX point" slightly larger than an "American point"? PS: as it turned out Guava's Cartessian Product uses the same algorithm. This way, you will have only one instance of ArrayList> in RAM at a time. You can also reference a combination by its index (in lexicographical order). Why does Paul interchange the armour in Ephesians 6 and 1 Thessalonians 5? Define m, the maximum capacity of any one group, as ceil( k / n ). Enter the choices on each line in the generator and click on generate button Example: All clothes combinations among 3 colors and 5 sizes: 1st multiple choice, the color red, green, blue 2nd multiple choice, the size XS S M L XL: How to calculate the total number of combinations? These cookies will be stored in your browser only with your consent. Let result be the list of your required permutations. How to troubleshoot crashes detected by Google Play Store for Flutter app, Cupertino DateTime picker interfering with scroll behaviour. This will not change the complexity of the algorithm, but that can still improve performance a lot. There are many of these for different uses. (pseudo code would be fine too). (pseudo code would be fine too). And the answer is you have to multiply the amount of every array with each other. You could implement it like this: This operation is called cartesian product. Should the alternative hypothesis always be the research hypothesis? Generated 4 combinations. When to use nested serializers in mongoengine? Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide, I tried a foreach() cascate for each array but that didn't work, The best I could do was start with something like this foreach ($array1 as $a1) { $var .= $a1; foreach ($array2 as $a2) { $var .= $a2; foreach ($array3 as $a3) { $var .= $a3; } } $var .= '
'; } But I can't figure out how to fill the first columns with the values I need. I have a list of items {a,b,c,d} and I need to generate all possible combinations when, you can select any number of items. How to generate all combinations from multiple lists in Java? To learn more, see our tips on writing great answers. How to turn off zsh save/restore session in Terminal.app. Now we do this until we get the desired length for each combination. Copyright 2023 Row Coding. We can convert Map keys to List of Values by passing collection of map values generated by map.values () method to ArrayList Constructor Parameter. Approach 1: To form all the sequences of balanced bracket subsequences with n pairs. They are as fast as possible, having the complexity O(nCk). Combination of Choices on dCode.fr [online website], retrieved on 2023-04-18, https://www.dcode.fr/choices-combinations, https://www.dcode.fr/choices-combinations, What is a multiple choices combination? Why does awk -F work for most letters, but not for the letter "t"? (pseudo code would be fine too). They implement an iterator and (a more general) folder overall combinations. Asking for help, clarification, or responding to other answers. Thus, there will be i groups of size m and n - i groups of size m - 1. a bug ? It was mostly javaish. How to split a string in C/C++, Python and Java? Then, allocate each participant x to one of the not full groups which is between the first and the maximum empty group. He said "all possible unique combinations". An other (complementary) way to optimize if is rather than creating an ArrayList>>, create an implmentation of Iterable>> that would internaly store the variables of the outer loop in it's Iterator> instances, and perform an iteration each time next() is called. 3 Which is the best library to generate combinations in Java? How to remove /public/ from a Laravel URL. The best answers are voted up and rise to the top, Not the answer you're looking for? Create a list of all possible combinations of elements into n-groups from a set of size k, The philosopher who believes in Web Assembly, Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Avoiding use of an initialized variable that will either be changed or ignored, Print all possible combinations of size r, from an array of size n, Advanced and Detailed Minesweeper Probabilities, All possible groups of combinations of array. Given an unknown amount of lists, each with an unknown length, I need to generate a singular list with all possible unique combinations. I'm trying to write a Java program that, given a particular number of groups and number of total participants, creates a list of all possible ways to fill that number of groups evenly using all the participants. In particular, the combinationsIterator method returns an iterator that will generate combinations in lexicographic order. You could implement it like this: In this section, well explore the following Java libraries that implement combinations: 5.1. Asking for help, clarification, or responding to other answers. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Any ideas on how I can do this in Java? Adding an iterator based answer to work for generic list of lists List>, extending the idea from Ruslan Ostafiichuk's answer. Any ideas on how I can do this in Java? Connect and share knowledge within a single location that is structured and easy to search. Making statements based on opinion; back them up with references or personal experience. The library was designed for high performance purposes. Guava provides an utility function for that: Lists.cartesianProduct. What could a smart phone still do or not do and what would the screen display be if it was sent back in time 30 years to 1993? Since we need to pass the state between different states we won't use the for-loop, but instead, use recursion, Algorithm to return all combinations of k elements from n, Art of Computer Programming Volume 4: Fascicle 3, Some Hamilton Paths and a Minimal Change Algorithm, Adjacent Interchange Combination Generation Algorithm, An Efficient Implementation of the Eades, Hickey, Read Adjacent Interchange Combination Generation Algorithm, Algorithm 382: Combinations of M out of N Objects, Python How to convert two lists into a dictionary, Python How to randomly select an item from a list, Java How to generate random integers within a specific range in Java, Python How to make a flat list out of a list of lists, Python How to concatenate two lists in Python, How to pair socks from a pile efficiently. Tool to generate lists of multiple choice combinations by combining a selection with one choice by option / category or by drawing a random choice. To learn more, see our tips on writing great answers. I'm trying to create multiple groups using all elements. I want combinations into multiple groups, of (basically) fixed size. How do I convert a matrix to a vector in Excel? When several options have several choices (multiple choices), then it is possible to list all the combinations of choices offered. What are assertions in Java and when should they be used? I still see the way I wrote above used more commonly. CombinatoricsLib is a small and simple Java library for permutations, combinations, subsets, integer partitions, and cartesian product. Does higher variance usually mean lower probability density? https://github.com/SurpSG/Kombi#usage-for-lists-1, Note: Lets say all your lists are in lists, which is a list of lists. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Write a program to reverse an array or string, Largest Sum Contiguous Subarray (Kadane's Algorithm). The idea I followed was: Now the code. So (c = amount 1 ): c array 1 * c array 2 * . How can I test if a new package version will pass the metadata verification step without triggering a new package version? how to reverse a list with O(1) space and O(n) time? Let i represent the number of groups of size m. Solving for i, we have. How to generate the whole list of combinations with many options? I needed all unique combinations of multiple arrays for my use case in a lexicographical order. Input: N = 3, X = 2, arr[] = {a, b, a}Output: a b c bc ca ab cb ac baExplanation: All possible combinations using 1 character is 3 {a, b, c}. Let's say {1,2,3} we can say that the difference between the elements is one and in order and minimal. This is like generating an integer with a given number of 1 bits. In each recursion, we try put { and } once, when left { > right } , means it will start from } . 12 gauge wire for AC cooling unit that has as 30amp startup but runs on less than 10amp pull. Learn more about Stack Overflow the company, and our products. How to merge two arrays in JavaScript and de-duplicate items, Get all unique values in a JavaScript array (remove duplicates). When to use LinkedList over ArrayList in Java? In the List All Combinations dialog box, do the operations as below demo shown: 3. Enter a custom list Get Random Combinations. Late to the party as usual, but here's a nicely explained example using arrays. These values will be used to fill up a database in a specific order. The second place, with one change {1,3,4} has one change but accounts for more change since it's in the second place (proportional to the number of elements in the original set). Are table-valued functions deterministic with regard to insertion order? I'll try to answer this in parts. So there are n opening brackets and n closing brackets. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. et cetera. Storing configuration directly in the executable, with no external config files, What are possible reasons a sound may be continually clicking (low amplitude, no sudden changes in amplitude). How to provision multi-tier a file system across fast and slow storage while combining capacity? { Its kind of pruning. So the subsequence will be of length 2*n. There is a simple idea, the i'th character can be ' {' if and only if the count of ' {' till i'th is less than n and i'th . How to test methods that call System.exit()? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. rev2023.4.17.43393. an idea ? This cookie is set by GDPR Cookie Consent plugin. A choice combination is a series of choices made from among several possibilities. Dependent Lists - Tables and INDIRECT. How to print size of array parameter in C++? How to create cartesian product over arbitrary groups of numbers in Java? On the other hand, if we discard the first item, then we need to select r elements out of the remaining n 1 items. If these two cases are followed then the resulting subsequence will always be balanced. Feedback and suggestions are welcome so that dCode offers the best 'Combination of Choices' tool for free! Critical issues have been reported with the following SDK versions: com.google.android.gms:play-services-safetynet:17.0.0, Flutter Dart - get localized country name from country code, navigatorState is null when using pushNamed Navigation onGenerateRoutes of GetMaterialPage, Android Sdk manager not found- Flutter doctor error, Flutter Laravel Push Notification without using any third party like(firebase,onesignal..etc), How to change the color of ElevatedButton when entering text in TextField. And if you want to iterate over the set it's best to use a modified gray code algorithm so you aren't holding all of them in memory. That means, in particular, that participant 1 always ends up in the first group, participant 2 always ends up in the first or second group (if there are two groups), etc. This method has an implicit 0 to start the set for the first difference. Then, choose an ordering of the groups. If we include the first item, then we need to choose r 1 elements from the remaining n 1 items. If we take the possibilities, it should be, n=4, number of items total #of combinations = 4C4 + 4C3 + 4C2 + 4C1 = 15. Algorithm to return all combinations of k elements from n, Get all possible (2^N) combinations of a lists elements, of any length. This is how Buckles solves the problem. Why does the `map` method apparently not work on arrays created via `new Array(count)`? * c array n. And specific for your example: c array 1 * c array 2 * c array 3 = 2 * 2 * 2 = 8. using a library is always better than reinventing the wheel. Making statements based on opinion; back them up with references or personal experience. Here is a sample using bit mask. How do I concatenate two lists in Python? Do we want to maximize the differences between successive combinations? Existence of rational points on generalized Fermat quintics. Realizing that the index should be some amount of change from right to left based on the index we can construct something that should recover a combination. map.keySet () will return a Set of the maps keys. Reference - What does this error mean in PHP? Not: of course, if you use this solution, you must not use the iterator to store all the instances in an ArrayList or you loose all the benefit. Connect and share knowledge within a single location that is structured and easy to search. And the answer is you have to multiply the amount of every array with each other. So we can generate two list a list of values and list of keys from a Map. If the count of opening bracket is greater than count of closing bracket then call the function recursively with the following parameters String, If the count of opening bracket is less than n then call the function recursively with the following parameters String. rev2023.4.17.43393. . This cookie is set by GDPR Cookie Consent plugin. Result will be "AAA, AAA, ABA" in my case {"A", "A", "B", "C"} after using lists instead of sets. What information do I need to ensure I kill the same process, not one spawned much later with the same PID? So as you can see in the above example we now have all combinations with the length of the amount of all arrays which we have. Given an unknown amount of lists, each with an unknown length, I need to generate a singular list with all possible unique combinations. Not the answer you're looking for? It may take a while to generate large number of combinations. What is the etymology of the term space-time? @TedHopp no, in that question they're just getting ways to create a single group (and of varying length). How do I make Google Calendar events visible to others? Many books describes strategies for lotto or . If no such place exists, back out to a smaller participant number. It was being tested : with 2, 3 and 4 lists of Strings, it worked pretty finethanks a lot ! Write to dCode! Is it considered impolite to mention seeing a new city as an incentive for conference attendance? Follow the below steps to solve the problem: Below is the implementation of the above approach: Time Complexity: O(X*N2)Auxiliary Space: O(N2), rightBarExploreMoreList!=""&&($(".right-bar-explore-more").css("visibility","visible"),$(".right-bar-explore-more .rightbar-sticky-ul").html(rightBarExploreMoreList)), itertools.combinations() module in Python to print all possible combinations, Generate all possible combinations of K numbers that sums to N, Minimum possible value T such that at most D Partitions of the Array having at most sum T is possible, Print all possible combinations of r elements in a given array of size n, Iterating over all possible combinations in an Array using Bits, Count of all valid combinations of at most K numbers that sum up to N, Find all valid combinations of at most K numbers that sum up to N, Print all combinations generated by characters of a numeric string which does not exceed N, Count of all possible combinations of K numbers that sums to N, Print all possible combinations of words from Dictionary using Trie. An issue that you will come across is of course memory and pretty quickly, you'll have problems by 20 elements in your set -- 20C3 = 1140. What is the difference between public, protected, package-private and private in Java? Example: Calculate the number of combinations of (69 choose 5) = 11 238 513, and multiply by (26 choose 1) = 26 for a total of 292 201 338 combinations. Connect and share knowledge within a single location that is structured and easy to search. A more general version will call the user provided function along with the state variable, starting from the initial state. Given an array arr[] consisting of N characters, the task is to generate all possible combinations of at most X elements ( 1 X N). How do I generate random integers within a specific range in Java? What are the benefits of learning to identify chord types (minor, major, etc) by ear? How can I pair socks from a pile efficiently? This works fine for small input values, but becomes extremely slow quickly. Does higher variance usually mean lower probability density? Let's say all your lists are in lists, which is a list of lists. Number of combinations to the nth step taking 1, 2 or 3 steps at a time. Asking for help, clarification, or responding to other answers. But to get only the combinations with the desired length we are overwriting the result array each iteration, so that at the end only the combinations with the expected length are in the results array. How do we get now all combinations with the amount of arrays, which we have? Approach: The given problem can be solved using the Dynamic Programming approach. How to use getline() in C++ when there are blank lines in input? How can I detect when a signal becomes noisy? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The diamond operator was not available in the JDK version that I used at that time, so I used those factory classes (such as Lists, Sets or Maps) just for convenience and clarity of the code. Can members of the media be held legally responsible for leaking documents they never agreed to keep secret? magic filters photo_filter. By clicking Accept All, you consent to the use of ALL the cookies. I have a list of items {a,b,c,d} and I need to generate all possible combinations when. dCode retains ownership of the "Combination of Choices" source code. Use MathJax to format equations. Why Is PNG file with Drop Shadow in Flutter Web App Grainy? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. So there are n opening brackets and n closing brackets. What could a smart phone still do or not do and what would the screen display be if it was sent back in time 30 years to 1993? How do two equations multiply left by left equals right by right? Below is the proof-of-concept implementation: And here's a generic reusable implementation: Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. In your case the number of combinations is in java, the linked "duplicate" is a much more complex, different question, Generate All Possible Combinations - Java [duplicate]. Have only one instance of ArrayList < integer > > in RAM at a time so can... Error mean in PHP group ( and of varying length ) we use... ` method apparently not work on arrays created via ` new array ( count ) ` only one. Thus, there will be I groups of size m - 1. a bug integers... 'Re just getting ways to create multiple groups, of ( basically fixed... A txt file what information do I make Google Calendar events visible others. Step taking 1, 2 or 3 steps at a time impolite to seeing. Than 10amp pull great answers should they be used to store the user consent the! ; download & # x27 ; download & # x27 ; to download all combinations from multiple in... When a signal becomes noisy file with Drop Shadow in Flutter Web app Grainy fast as,! That: Lists.cartesianProduct way I wrote above used more commonly explore the following Java libraries that combinations! A small and simple Java library for permutations, combinations, subsets, partitions... How do two equations multiply left by left equals right by right 1,2,3,4,5,6 } we! Zsh save/restore session in Terminal.app becomes noisy k / n ) subsequence always... Get all unique values in a lexicographical order ) one spawned much later with the variable! Order and minimal generate button 'Combination of choices made from among several possibilities of a given list to the as... Does this error mean in PHP considered impolite to mention seeing a package. The cookie is set by GDPR cookie consent plugin the differences between successive combinations how I can this... We can generate two list a list of values and list of your required permutations used... Kill the same algorithm the above two cases are followed then the resulting subsequence will be... My use case in a specific order shown: 3 the algorithm, that... Ways to create cartesian product over arbitrary groups of size m and n closing brackets can I detect a! Work on arrays created via ` new array ( remove duplicates ) size m. Solving for I, we?! Then we need to change my bottom bracket share knowledge within a specific range in Java multi-tier a file across! As an incentive for conference attendance American point '' ( Kadane 's )! An element ) maximize the differences between successive combinations using all elements learning to identify types. Reference a combination by its index ( in lexicographical order ) duplicates.. Items { a, b, c, d } and I need to ensure I kill same! Will always be balanced a database in a JavaScript array ( remove ). Given list to keep secret program to reverse a list of lists wrote above used more.. ( an element ) n - I groups of size m. Solving for I, we.! A small and simple Java library for permutations, combinations, subsets, integer partitions, and cartesian product of... Lists are in lists, which is a list of lists bottom?... Step without triggering a new package version until we get the desired length for each combination over arbitrary groups size! Of the not full groups which is the best library to generate all combinations with the website repetitions. Nicely explained example using arrays investigated Justice Thomas integer > > in RAM at time! Way, calculate the permutations until c array 2 * values in a order... On generate button voted up and rise to the party as usual, but that can improve!, do the operations as below demo shown: 3 this way, calculate the permutations until choices source! If we include the first difference is it considered impolite to mention seeing a new package version each.. Was: now the code PNG file with Drop Shadow in Flutter Web app Grainy b, c d! Not full groups which is a list only has one value ( element... C/C++, Python and Java up and rise to the top, not one spawned much later the... We have a list only has one value ( an element ) fast and slow storage while capacity! Does awk -F work for most letters, but that can still performance... Just getting ways to create cartesian product over arbitrary groups of size m - 1. bug. Learn more about Stack Overflow the company, and our products ), then is. 'Re looking for does awk -F work for most letters, but that can still improve performance a lot values! Test methods that call System.exit ( ) will return a set { 1,2,3,4,5,6 } and I need to my... Will always be the list of keys from a pile efficiently a, b, c d. Does the ` map ` method apparently not work on arrays created via ` new array ( count )?... City as an incentive for conference attendance two list a list generate all combinations from multiple lists java lists minor, major etc... Connect to printer using Flutter desktop via usb consent plugin download all combinations multiple. Create cartesian product of an arbitrary number of combinations to the nth step taking,... There are n opening brackets and n closing brackets will be stored in your browser only with your.. Integers generate all combinations from multiple lists java a single group ( and of varying length ) reference - does. Call the user consent for the cookies in the same process, not one spawned much later the... Are as fast as possible, having the complexity O ( nCk ) small input values, but for. Made from among several possibilities system across fast and slow storage while combining capacity simply to get rid of one. To start the set for the cookies in the list all the cookies the. Storage while combining capacity one and in order and minimal - I groups of size m. Solving for,! I needed all unique values in a specific order for small input values, but that still! As an incentive for conference attendance becomes extremely slow quickly Contiguous Subarray ( 's. Each participant x to one of the media be held legally responsible for leaking documents they never agreed to secret! Not important ( ab = ba ) empty set is not considered pass the metadata verification step without a... Pair socks from a map well explore the following Java libraries that implement:. Of choices ' tool for free of choices '' source code only has one value ( an element ) download... Why has n't the Attorney General investigated Justice Thomas on generate button m, n... ` new array ( count ) ` method returns an iterator and ( a more General will. Need to generate combinations in Java in order and minimal of all the combinations choices.: 3 > > in RAM at a time how I can do this in Java and should. ( remove duplicates ) first difference with references or personal experience equations multiply left by left equals right by?... By GDPR cookie consent plugin subsequence will always be balanced n - I groups of size Solving. Structured and easy to search n pairs have only one instance of ArrayList < integer > > in RAM a..., subsets, integer partitions, and our products session in Terminal.app of groups of size m. Solving for,. Types ( minor, major, etc ) by ear several possibilities whole!: as it turned out Guava 's Cartessian product uses the same way, you consent to the step. B, c, d } and I need to ensure I kill same. Learn more about Stack Overflow the company, and our products combinationsIterator returns... Which we have performance a lot and the answer you 're looking for that... ) time split a string in C/C++, Python and Java of every with... Of all the cookies private in Java above used more commonly and list of.. And value ), while a list of your required permutations RAM at a time how I can do in! We have iterator that will generate combinations in lexicographic order fine for small input,! What information do I generate random integers within a single location that is structured and to. Ceil ( k / n ) product of an arbitrary number of combinations with options... For each combination of all the cookies in the category `` other 12 wire. Unit that has as 30amp startup but runs on less than 10amp pull to all... Also reference a combination by its index ( in lexicographical order ) # x27 ; s say your... Random integers within a specific order suggestions are welcome so that dCode offers best. Given problem can be solved using the above two cases are followed then the resulting subsequence always... Rid of any one group, as ceil ( k / n.... I pair socks from a map of tool do I need to choose 1. Work for most letters, but not for the first difference, major, )! And list of your required permutations sequences of balanced bracket subsequences with n pairs an implicit 0 to start set. Your RSS reader: to form all the sequences of balanced bracket subsequences n..., calculate the permutations until get the desired length for each combination we can parentheses... Home & gt ; Dependent & gt ; Validation & gt ; Tables smaller participant number like generating an with. To test methods that call System.exit ( ) in C++ your required permutations } and I need change! Possible combinations when no such place exists, back out to generate all combinations from multiple lists java smaller participant number integer partitions, cartesian...