From 1af779b077223a175544d8910e7ec023b817340f Mon Sep 17 00:00:00 2001 From: hearot Date: Wed, 12 Nov 2025 10:24:51 +0100 Subject: [PATCH] combalg: aggiunge formulario preliminare --- .../Formula sheet/formulae.tex | 110 ++++++++++++++++++ .../Formula sheet/main.pdf | Bin 0 -> 147084 bytes .../Formula sheet/main.tex | 17 +++ .../Formula sheet/preamble.tex | 45 +++++++ 4 files changed, 172 insertions(+) create mode 100644 Corsi/Combinatoria algebrica/Formula sheet/formulae.tex create mode 100644 Corsi/Combinatoria algebrica/Formula sheet/main.pdf create mode 100644 Corsi/Combinatoria algebrica/Formula sheet/main.tex create mode 100644 Corsi/Combinatoria algebrica/Formula sheet/preamble.tex diff --git a/Corsi/Combinatoria algebrica/Formula sheet/formulae.tex b/Corsi/Combinatoria algebrica/Formula sheet/formulae.tex new file mode 100644 index 0000000..8f21437 --- /dev/null +++ b/Corsi/Combinatoria algebrica/Formula sheet/formulae.tex @@ -0,0 +1,110 @@ +\begin{formula}[Sum of two power series] + \[ \sum_{i \geq 0} a_i x^i + \sum_{i \geq 0} b_i x^i \triangleq \sum_{i \geq 0} (a_i + b_i) x^i. \] +\end{formula} + +\begin{formula}[Sum of multiple power series] + \[ \bigplus_{i \in [n]} \sum_{j \geq 0} a_{i, j} \, x^j = \sum_{j \geq 0} \left(\sum_{i \in [n]} a_{i, j}\right) x^j. \] +\end{formula} + +\begin{formula}[Product of two power series] + \[ \left( \sum_{i \geq 0} a_i x^i \right) \cdot \left( \sum_{i \geq 0} b_i x^i \right) \triangleq \sum_{i \geq 0} \left( \sum_{j = 0}^i a_j b_{i-j} \right) x^i. \] +\end{formula} + +\begin{formula}[Product of multiple power series] + \[ \prod_{i \in [n]} \sum_{j \geq 0} a_{i, j} \, x^j = \sum_{j \geq 0} \left( \sum_{k_1 + \ldots + k_n = j} a_{1, k_1} \cdots a_{n, k_n} \right) x^j. \] +\end{formula} + +\begin{formula}[Geometric series] + \[ \frac{x^k}{(1-x)} = \sum_{i \geq k} x^i \in \CC[[x]]. \] + + Prove the formula for $k = 1$ and then group $x$'s to retrieve the + general formula. +\end{formula} + +\begin{formula}[Exponential series] + \[ e^x \triangleq \sum_{i \geq 0} \frac{x^i}{i!}. \] +\end{formula} + +\begin{formula}[Logarithmic series] + \[ \log(1+x) \triangleq \sum_{i \geq 1} (-1)^{i+1} \, \frac{x^i}{i}. \] + + It's ``$\int \nicefrac{1}{(1+x)} \dx$''. +\end{formula} + +\begin{formula} + \[ \log\left(\frac{1}{1-x}\right) = - \log(1-x) = \sum_{i \geq 1} \frac{x^i}{i}. \] +\end{formula} + +\begin{formula}[Binomial coefficient] + \[ \binom{n}{k} \triangleq \# \{B \subseteq [n] \mid \abs{B} = k\}. \] + + Number of ways for choosing $k$ elements out of $n$. +\end{formula} + +\begin{formula} \label{fm:binomial_recursion} + \[ \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}, \quad n \geq 1. \] + + You either choose $n$ or you don't. +\end{formula} + +\begin{formula}[Newton's binomial theorem] \label{fm:newton} + \[ (1+x)^n = \sum_{k=0}^n \binom{n}{k} x^k \] + + Apply \autoref{fm:binomial_recursion} using induction. +\end{formula} + +\begin{formula} + \[ \#\{B \subseteq [n]\} = 2^n. \] + + Apply \autoref{fm:newton} with $x = 1$. Alternatively, + each subset $B$ is uniquely identified by + its characteristics function $1_B$, hence the subsets of $[n]$ + are counted by the functions from $[n]$ to $[2]$. +\end{formula} + +\begin{formula} + \[ (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}. \] + + Apply \autoref{fm:newton}. +\end{formula} + +\begin{formula}[Formula for the binomial coefficient] + \[ \binom{n}{k} = \frac{n!}{(n-k)! k!}. \] + + Apply \autoref{fm:newton} and derive $(1+x)^n$ $k$ times. +\end{formula} + +\begin{formula}[Falling factorials] + \[ (x)_k \triangleq x(x-1) \cdots (x-k+1) = \prod_{i=0}^{k-1} (x-i), \quad x \in \CC. \] +\end{formula} + +\begin{formula}[Rising factorials] + \[ x^{(k)} \triangleq x(x+1) \cdots (x+k-1) = \prod_{i=0}^{k-1} (x+i), \quad x \in \CC. \] +\end{formula} + +\begin{formula}[Binomial coefficients with $n \in \CC$] + \[ \binom{n}{k} \triangleq \frac{(n)_k}{k!}, \quad n \in \CC, k \in \NN. \] + + This is compatible with how binomials + were previously defined. +\end{formula} + +\begin{formula}[Newton's binomial theorem for falling factorials] \label{fm:newton_falling} + \[ (a+b)_n = \sum_{i=0}^n (a)_i (b)_{n-i}. \] + + By induction on $n$. +\end{formula} + +\begin{formula}[Order of a formal power series] + \[ \ord(f(x)) \triangleq \mdeg(f(x)) \triangleq \min \{ i \mid a_i \neq 0 \}. \] +\end{formula} + +\begin{formula}[Existence of $k$-roots in $\CC$-power series] + $f(x)$ admits a $k$-root in $\CC[[x]]$ if and only if $k \mid \ord(f(x))$. +\end{formula} + +\begin{formula}[$k$-roots of $(1+x)$] + \[ (1+x)^{\nicefrac{1}{k}} = \sum_{i\geq 0} \binom{\nicefrac{1}{k}}{i} x^i. \] + + Apply \autoref{fm:newton_falling}). +\end{formula} diff --git a/Corsi/Combinatoria algebrica/Formula sheet/main.pdf b/Corsi/Combinatoria algebrica/Formula sheet/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b026c5760991d8c6c1ea6c7dc255565b5a644c0f GIT binary patch literal 147084 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zeLr5%vv5gxb~uKOG}O_wvslzUC?@+B^`1cEgI3v7!>aJi0XqT*><9JRzK1Pr`TfOf zD&LWZw(57B^1Q-wm?|4Ko2y@i+w_(9B{{LIo35{Zhs68-6zF*&?DfQg{wuEHpT&s3 zkFRyrEv-;Elj7q9asj2KrNp?n__-ydc=)8nfa1K8yqr9oKw+wX-vn|W zMN3C(2m`{+`JYcsqnmom(W^!jbbk9>{@(UlytK^DQ2r>O`1IB;MY3}P;lU4guDHWpMjr-hEApr+hbZpoC&0gz|I(nfRDU@4b-VajV4{?+g$Gj3pnVv*^=FG*xSbB?~6))Zxp^4(u7@<|=+9<6_A=D_X zWz^;Hp)|S5;X`@V%b9Y?Zha%Y7RRZ*QmIkDy;8Yw=2E&E8-cy0rbv$cSlP?34C559 zP!=m^O44@k(6do25HQRMEs;>-suxRzNUswu+FF9kF<~iOXz;u6_Bs^CPr23fjnU{Q k=)O)Cmw)|fxVf9Ux`QEv6AFNbkC%%Jg^o^2MH=P50Es>HuK)l5 literal 0 HcmV?d00001 diff --git a/Corsi/Combinatoria algebrica/Formula sheet/main.tex b/Corsi/Combinatoria algebrica/Formula sheet/main.tex new file mode 100644 index 0000000..8c42d72 --- /dev/null +++ b/Corsi/Combinatoria algebrica/Formula sheet/main.tex @@ -0,0 +1,17 @@ +\documentclass[10pt]{report} +\input{preamble.tex} + +\raggedcolumns +\begin{document} + +\begin{center} + \Large \textbf{Formula sheet for Algebraic combinatorics} +\end{center} + +\setlength{\columnseprule}{0.1pt} +\setlength{\columnsep}{25pt} + +\begin{multicols*}{2} + \input{formulae.tex} +\end{multicols*} +\end{document} diff --git a/Corsi/Combinatoria algebrica/Formula sheet/preamble.tex b/Corsi/Combinatoria algebrica/Formula sheet/preamble.tex new file mode 100644 index 0000000..e08b090 --- /dev/null +++ b/Corsi/Combinatoria algebrica/Formula sheet/preamble.tex @@ -0,0 +1,45 @@ +\usepackage[top=1.5cm,bottom=1.5cm,left=1.5cm,right=1.5cm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{enumerate} +\usepackage{hyperref} +\usepackage{mathtools} +\usepackage{multicol} +\usepackage{nicefrac} +\usepackage{relsize} +\usepackage{stmaryrd} + +\makeatletter +\newtheoremstyle{plaintext} +{20pt} +{20pt} +{\normalfont} +{} +{\bfseries} +{.} +{\newline} +{\thmname{#1}\thmnumber{ #2}\thmnote{ (#3)}} +\makeatother +\theoremstyle{plaintext} + +\newtheorem{formula}{Formula} +\providecommand*{\formulaautorefname}{Formula} + +\newcommand{\NN}{\mathbb{N}} +\newcommand{\ZZ}{\mathbb{Z}} +\newcommand{\QQ}{\mathbb{Q}} +\newcommand{\RR}{\mathbb{R}} +\newcommand{\CC}{\mathbb{C}} + +\newcommand{\inv}{^{-1}} +\newcommand{\abs}[1]{\left\lvert #1 \right\rvert} +\newcommand{\dx}{\, \mathrm{d}x} + +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\mdeg}{mdeg} +\DeclareMathOperator*{\bigplus}{\mathlarger{\mathlarger{\mathlarger{+}}}} + +\allowdisplaybreaks