An Invitation to Combinatorics

Active student engagement is key to this classroom-tested combinatorics text, boasting 1200+ carefully designed problems, ten mini-projects, section warm-up problems, and chapter opening problems. The author – an award-winning teacher – writes in a conversational style, keeping the reader in mind on every page. Students will stay motivated through glimpses into current research trends and open problems as well as the history and global origins of the subject. All essential topics are covered, including Ramsey theory, enumerative combinatorics including Stirling numbers, partitions of integers, the inclusion-exclusion principle, generating functions, introductory graph theory, and partially ordered sets. Some significant results are presented as sets of guided problems, leading readers to discover them on their own. More than 140 problems have complete solutions and over 250 have hints in the back, making this book ideal for self-study. Ideal for a one semester upper undergraduate course, prerequisites include the calculus sequence and familiarity with proofs.

k Part Partitions of Integers

This paper presents formulas (together with their proofs) determining 3 and 4 part partitions of any integer. These formulas were derived using the properties of the floor function and Bernoulli formulas for various powers of finite sums of the floor function series. This made it possible to obtain above formulas in a much simpler way than most traditional methods.

On relations for the partitions of numbers

In this paper, we present some interrelations among some restricted and unrestricted partitions of integers. Mainly, we derive new effective formulas for the partition function and compare our partition formula with well known recurrence formulas. Moreover, as the number increase, we observe how effective the new recurance formulas are.

Visual thinking and simplicity of proof

This paper studies how spatial thinking interacts with simplicity in [informal] proof, by analysing a set of example proofs mainly concerned with Ferrers diagrams (visual representations of partitions of integers) and comparing them to proofs that do not use spatial thinking. The analysis shows that using diagrams and spatial thinking can contribute to simplicity by (for example) avoiding technical calculations, division into cases, and induction, and creating a more surveyable and explanatory proof (both of which are connected to simplicity). In response to one part of Hilbert's 24th problem, the area between two proofs is explored in one example, showing that between a proof that uses spatial reasoning and one that does not, there is a proof that is less simple yet more impure than either. This has implications for the supposed simplicity of impure proofs. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

Profil number sense siswa bergaya kognitif visualizer terhadap makna pecahan desimal

[Bahasa]: Penelitian kualitatif ini bertujuan untuk mendeskripsikan profil number sense siswa terhadap makna pecahan desimal. Subjek penelitian adalah satu siswa kelas VII SMP Bhayangkari Kemala I Surabaya dengan kemampuan matematika tinggi dan bergaya kognitif visualizer. Penelitian dimulai dengan menentukan subjek penelitian mengunakan instrumen tes gaya kognitif dan tes kemampuan matematika, kemudian dilanjutkan dengan pemberian tes number sense (TNS). Tahap terakhir adalah melakukan wawancara dengan subjek untuk mengungkap cara berfikir siswa dalam menyelesaikan soal tes number sense serta melihat kesesuaian jawaban dengan alasan yang diberikan. Pengecekan keabsahan data dalam penelitian ini menggunakan triangulasi waktu. Hasil penelitian menunjukkan bahwa number sense yang dimiliki oleh subjek dalam memahami makna dasar pecahan desimal ditunjukkan dengan mempresentasikan pecahan desimal sebagai pecahan biasa, partisi dari bilangan bulat, dan partisi suatu benda. Pemahaman mengenai urutan pecahan desimal ditunjukkan dengan meletakkan pecahan-pecahan desimal pada garis bilangan sesuai urutan yang benar. Pemahaman mengenai sifat kerapatan pecahan desimal ditunjukkan dengan penyimpulan bahwa ada tidak hingga pecahan desimal antara dua pecahan desimal. Jadi, dapat disimpulkan bahwa siswa yang bergaya kognitif visualizer dengan kemampuan matematika tinggi dapat memahami makna pecahan desimal. Kata kunci: Number Sense; Gaya Kognitif; Visualizer; Pecahan Desimal [English]: This qualitative research aimed to describe the profile of student’s number sense toward the meaning of decimal. The research subject was one 7th grade of SMP Bhayangkari Kemala I Surabaya with high mathematics achievement and visualizer cognitive-style. The research began by determining the subject using cognitive-style test instrument and mathematics tests, then followed by the number sense test (TNS). The last stage was interviewing the subject to reveal how the subject think in solving the number sense test and examine the match between the answers and the reasons given. Time triangulation was used to check the validity of data. The research found that the number sense possessed by the subject in understanding the basic meaning of decimal is representing decimal fractions as regular fractions, partitions of integers, and partitions of an object. Understanding of the order of decimal is shown by placing the decimal on the number line in the correct order. Understanding of the nature of the decimal density is denoted by the conclusion that there are infinite decimals between two decimals. Thus, it could be concluded that students with visualizer cognitive-style and high mathematics achievement can understand the meaning of decimal properly. Keywords : Number Sense; Cognitive Style; Visualizer; Decimal