ocaml: phase 5.1 count_palindromes.ml baseline ("aabaa" -> 9 palindromes)

Expand-around-center linear-time palindrome counting:

  for c = 0 to 2*n - 2 do
    let l = ref (c / 2) in
    let r = ref ((c + 1) / 2) in
    while !l >= 0 && !r < n && s.[!l] = s.[!r] do
      count := !count + 1; l := !l - 1; r := !r + 1
    done
  done

The 2n-1 centers cover both odd (c even -> l = r) and even
(c odd -> l = r - 1) palindromes.

For "aabaa":
  5 singletons + 2 "aa" + 1 "aba" + 1 "aabaa" = 9

Complements lps_dp.ml (longest subsequence) and manacher.ml
(longest substring); this one *counts* all palindromic substrings.

187 baseline programs total.

plans/ocaml-on-sx.md

@@ -407,6 +407,15 @@ _Newest first._
   binary search tree (`type 'a tree = Leaf | Node of 'a * 'a tree *
   'a tree`) with insert + in-order traversal. Tests parametric ADT,
   recursive match, List.append, List.fold_left.
+- 2026-05-11 Phase 5.1 — count_palindromes.ml baseline (count
+  palindromic substrings of "aabaa" = 9). Expand-around-center
+  with 2n−1 centers (n odd-length, n−1 even-length); for each,
+  walk outward while characters match and both indices in range.
+  Decomposition: c/2 and (c+1)/2 gives both odd (c even) and even
+  (c odd) starting pairs. Palindromes in "aabaa": 5 singletons
+  + 2 "aa"s + 1 "aba" + 1 "aabaa" = 9. Complements lps_dp.ml
+  (longest subsequence) and manacher.ml (longest substring); this
+  one counts ALL palindromic substrings. 187 baseline programs total.
 - 2026-05-11 Phase 5.1 — count_subarrays_k.ml baseline (count
   contiguous subarrays of [1;1;1;2;-1;3;1;-2;4] summing to k=3
   = 7). Prefix-sum brute force in O(n²): build cumulative array,