Catalan Numbers

algorithms/catalan.typ

@@ -0,0 +1,89 @@
+#import "../lib/style.typ": *
+#import "../lib/mapcode.typ": *
+
+== Catalan Numbers
+#set math.equation(numbering: none)
+
+Compute the $n$-th Catalan number.
+Formal definition:
+$
+C_0 &= 1\
+C_{n+1} &= sum_(i=0)^n C_i C_{n-i} quad "for " n >= 0
+$
+
+Examples:
+- $"Cat"(0) -> 1$
+- $"Cat"(3) -> 5$
+- $"Cat"(6) -> 132$
+
+*As mapcode:*
+
+_primitives_: `sum`($+$), `product`($*$)
+
+$ I = n:NN quad quad quad X_n &= [0..n] -> NN_bot quad quad quad A = NN\
+rho(n) & = {i -> bot | i in {0 dots n}}\
+F(n)(x) & = cases(
+  1 & "if " k = 0,
+  sum_(i=0)^(k-1) x[i] dot x[k-1-i] & "if " k > 0 and forall i: x[i] != bot,
+  bot & "otherwise"
+ )\
+ pi(n)(x) & = x[n]
+$
+
+#let inst = 6;
+#figure(
+  caption: [Catalan number computation using mapcode for $n = #inst$],
+$
+#{
+  let rho = (inst) => {
+    let x = ()
+    for i in range(0, inst + 1) {
+      x.push(none)
+    }
+    x
+  }
+
+  let F_i = (x) => ((k,)) => {
+    if k == 0 {1}
+    else {
+        let sum = 0
+        let possible = true
+        for i in range(0, k) {
+            let left = x.at(i)
+            let right = x.at(k - 1 - i)
+            if left != none and right != none {
+                sum += left * right
+            } else {
+                possible = false
+                break
+            }
+        }
+        if possible { sum } else { none }
+    }
+  }
+  let F = map_tensor(F_i, dim: 1)
+
+  let pi = (inst) => (x) => x.at(inst)
+
+  let X_h = (x, diff_mask: none) => {
+    let cells = x.enumerate().map(((i, x_i)) => {
+      let val = if x_i != none {[$#x_i$]} else {[$bot$]}
+      if diff_mask != none and diff_mask.at(i) {
+        rect(fill: yellow.transparentize(70%), inset: 2pt)[$#val$]
+      } else {
+        rect(stroke: none, inset: 2pt)[$#val$]
+      }
+    })
+    $vec(delim: "[", ..cells)$
+  }
+
+  mapcode-viz(
+    rho, F, pi(inst),
+    X_h: X_h,
+    pi_name: [$mpi (inst)$],
+    group-size: calc.min(7, inst + 1),
+    cell-size: 10mm, scale-fig: 85%
+  )(inst)
+}
+$
+)

main.typ

@@ -52,3 +52,5 @@ All primitives are _strict_ meaning they do not allow for undefined values (i.e.
 #include "algorithms/LongestCommonSubsequence.typ"
 #pagebreak()
 #include "algorithms/leetcode/P2_add-two-numbers.typ"
+#pagebreak()
+#include "algorithms/catalan.typ"