Combinatorial combinations

bibliography/books.bib

@@ -1693,16 +1693,6 @@ @book{Архангельский1988Множества
   title = {Канторовская теория множеств}
 }
 
-@book{БелоусовВласов2012Комбинаторика,
-  author = {Алексей Иванович Белоусов and Павел Александрович Власов},
-  date = {2012},
-  language = {russian},
-  publisher = {Издательство МГТУ},
-  shortauthor = {Belousov and Vlаsov},
-  subtitle = {Методические указания к выполнению домашнего задания},
-  title = {Элементы комбинаторики}
-}
-
 @book{БелоусовТкачёв2004ДискретнаяМатематика,
   author = {Алексей Белоусов and Сергей Ткачёв},
   date = {2004},

figures/con__pascals_triangle.asy

@@ -1,32 +1,20 @@
-unitsize(3.5cm);
-
-pair A = (1, sqrt(3));
-pair B = (0, 0);
-pair C = (2, 0);
+unitsize(1cm);
 
 int n = 7;
 int[][] values = new int[n][n];
 
-for (int k = 0; k < n; ++k) {
-  draw(B + k / n * (A - B) -- C + k / n * (A - C), mediumgray);
-  draw(A + k / n * (B - A) -- C + k / n * (B - C), mediumgray);
-  draw(A + k / n * (C - A) -- B + k / n * (C - B), mediumgray);
-}
+pair o = (0, 0);
+pair u = unit((1, -sqrt(3)));
+pair v = unit((1, sqrt(3)));
 
 for (int i = 0; i < n; ++i) {
   for (int j = 0; i + j < n; ++j) {
-    pair translation = i / n * (B - A) + j / n * (C - A);
-    pair A_ = A + translation;
-    pair B_ = B + (n - 1) / n * (A - B) + translation;
-    pair C_ = C + (n - 1) / n * (A - C) + translation;
-    pair pos = (A_ + B_ + C_) / 3;
-
     if (i == 0 || j == 0) {
       values[i][j] = 1;
     } else {
       values[i][j] = values[i - 1][j] + values[i][j - 1];
     }
 
-    label(string(values[i][j]), pos);
+    label(string(values[i][j]), i * u - j * v);
   }
 }

figures/ex__def__triangular_point_configuration__tetractys.asy

@@ -1,7 +1,5 @@
 unitsize(1.3cm);
 
-from notebook access PointLattice;
-
 pair o = (0, 0);
 pair u = (1, 0) / 2;
 pair v = (1, -sqrt(3)) / 4;

text/field_extensions.tex

@@ -570,7 +570,7 @@ \section{Field extensions}\label{sec:field_extensions}
     It is also the minimal polynomial of \( \zeta + 1 \) because
     \begin{equation*}
       (\zeta + 1)^2 + \zeta + \underbrace{1 + 1}_0
-      \reloset {\eqref{eq:thm:newtons_binomial_theorem_positive_characteristic}} =
+      \reloset {\eqref{eq:thm:binomial_theorem_positive_characteristic}} =
       (\zeta^2 + 1) + \zeta
       \reloset {\eqref{eq:ex:def:finite_field/f4/minimal/characteristic}} =
       0.

text/figurate_numbers.tex

@@ -410,7 +410,7 @@ \section{Figurate numbers}\label{sec:figurate_numbers}
 
     \thmitem{ex:def:fermats_equation/complex} \Fullref{thm:fundamental_theorem_of_algebra} implies that the equation \( X^n - (a^n + b^n) \) has a solution for any pair \( (a, b) \) of complex numbers.
 
-    \thmitem{ex:def:fermats_equation/finite} Over the \hyperref[def:finite_field]{finite field} \( \BbbF_q \), \fullref{thm:newtons_binomial_theorem_positive_characteristic} implies that
+    \thmitem{ex:def:fermats_equation/finite} Over the \hyperref[def:finite_field]{finite field} \( \BbbF_q \), \fullref{thm:binomial_theorem_positive_characteristic} implies that
     \begin{equation*}
       a^q + b^q = (a + b)^q.
     \end{equation*}

text/lattice_fixed_points.tex

@@ -93,6 +93,9 @@ \section{Lattice fixed points}\label{sec:lattice_fixed_points}
 \begin{definition}\label{def:scott_continuity}\mcite[56]{Grätzer2011Lattices}
   We say that a map between \hyperref[def:complete_lattice]{complete lattices} is \term{Scott-continuous} if it preserves joins of \hyperref[def:directed_set]{upward-directed families}.
 \end{definition}
+\begin{comments}
+  \item Instead of complete lattices, we can work over \hyperref[def:partially_ordered_set]{partially ordered sets} closed under upward-directed suprema. \incite[def. 8.1]{DaveyPriestley2002Lattices} call such a partially ordered sets \enquote{complete}, which unfortunately conflicts with our notion of complete lattice. \incite[192]{Birkhoff1967Lattices} instead requires all \hyperref[def:partial_order_chain/chain]{chains} to have a supremum and calls such partially ordered sets \enquote{inductive}. \incite[83]{БелоусовТкачёв2004ДискретнаяМатематика} calls a partially ordered set \enquote{индуктивным} (\enquote{inductive}) if it satisfies Davey and Priestley's condition.
+\end{comments}
 
 \begin{proposition}\label{thm:def:scott_continuity}
   \hyperref[def:scott_continuity]{Scott-continuous} maps have the following basic properties:

text/primitive_roots.tex

@@ -55,12 +55,12 @@ \section{Primitive roots}\label{sec:primitive_elements}
   Then \( \BbbK \) contains \( m \) distinct \( n \)-th roots of unity, each with multiplicity \( p^e \).
 \end{lemma}
 \begin{proof}
-  \Fullref{thm:newtons_binomial_theorem_positive_characteristic} implies that
+  \Fullref{thm:binomial_theorem_positive_characteristic} implies that
   \begin{equation}\label{eq:thm:roots_of_unity_multiplicity/proof}
     X^n - 1
     =
     X^{m \cdot p^e} - 1^{p^e}
-    \reloset {\eqref{eq:thm:newtons_binomial_theorem_positive_characteristic}} =
+    \reloset {\eqref{eq:thm:binomial_theorem_positive_characteristic}} =
     (X^m - 1)^{p^e}.
   \end{equation}
 
@@ -589,7 +589,7 @@ \section{Primitive roots}\label{sec:primitive_elements}
     \sum_{i=0}^n a_i^{q^k} \alpha^{i q^k}
     =
     \sum_{i=0}^n (a_i X^i)^{q^k}
-    \reloset {\eqref{eq:thm:newtons_binomial_theorem_positive_characteristic}} =
+    \reloset {\eqref{eq:thm:binomial_theorem_positive_characteristic}} =
     \parens[\Big]{ \sum_{i=0}^n a_i X^i }^{q^k}
     =
     f(\alpha)^{q^k}
@@ -619,7 +619,7 @@ \section{Primitive roots}\label{sec:primitive_elements}
   Indeed, if \( i > 0 \), we have
   \begin{equation*}
     \parens[\Big]{ \alpha^{q^{m-1}} - \alpha^{q^{i-1}} }^q
-    \reloset {\eqref{eq:thm:newtons_binomial_theorem_positive_characteristic}} =
+    \reloset {\eqref{eq:thm:binomial_theorem_positive_characteristic}} =
     \alpha^{q^m} - \alpha^{q^i}
     =
     0,
@@ -645,7 +645,7 @@ \section{Primitive roots}\label{sec:primitive_elements}
     g(X)^q
     =
     \prod_{i=0}^{m-1} (X - \alpha^{q^i})^q
-    \reloset {\eqref{eq:thm:newtons_binomial_theorem_positive_characteristic}} =
+    \reloset {\eqref{eq:thm:binomial_theorem_positive_characteristic}} =
     \prod_{i=0}^{m-1} (X^q - \alpha^{q^{i + 1}})
     =
     \prod_{i=1}^m (X^q - \alpha^{q^i})
@@ -659,7 +659,7 @@ \section{Primitive roots}\label{sec:primitive_elements}
     g(X)^q
     =
     \parens[\Big]{ \sum_{i=0}^m b_i X^i }^q
-    \reloset {\eqref{eq:thm:newtons_binomial_theorem_positive_characteristic}} =
+    \reloset {\eqref{eq:thm:binomial_theorem_positive_characteristic}} =
     \sum_{i=0}^m b_i^q X^{qi}.
   \end{equation*}
 
@@ -759,7 +759,7 @@ \section{Primitive roots}\label{sec:primitive_elements}
       \alpha^8
       \reloset {\eqref{ex:def:finite_field_primitive_polynomial/f2_cubic/alpha_fourth}} =
       (\alpha^2 + \alpha + 1)^2
-      \reloset {\eqref{eq:thm:newtons_binomial_theorem_positive_characteristic}} =
+      \reloset {\eqref{eq:thm:binomial_theorem_positive_characteristic}} =
       \alpha^4 + \alpha^2 + 1
       \reloset {\eqref{ex:def:finite_field_primitive_polynomial/f2_cubic/alpha_fourth}} =
       \cancel{\alpha^2} + \alpha + \cancel{1} + \cancel{\alpha^2} + \cancel{1}