bibkey: colmenarejo2024lucky authors: Laura Colmenarejo, Aleyah Dawkins, Jennifer Elder, Pamela E. Harris, Kimberly J. Harry, Selvi Kara, Dorian Smith, Bridget Eileen Tenner year: 2024 title: “On the lucky and displacement statistics of Stirling permutations” doi: 10.48550/arXiv.2403.03280 url: https://cs.uwaterloo.ca/journals/JIS/VOL27/Tenner/tenner12.html claim: “Corollary 41 bounds the number of nonzero entries of a displacement composition between n and 2n-1, and Problem 48 asks whether every count in that range is attained.” strata_touched:

• D5/S3/Combinatorics/StirlingLuckyDisplacementSpectrum license: citation-only triage: anchor

Colmenarejo–Dawkins–Elder–Harris–Harry–Kara–Smith–Tenner lucky and displacement statistics

Verified locator

URL: https://cs.uwaterloo.ca/journals/JIS/VOL27/Tenner/tenner12.html

DOI: 10.48550/arXiv.2403.03280

Version: Journal of Integer Sequences 27 (2024), Article 24.6.7. The preprint arXiv:2403.03280v1, which has no later version, carries the same text but numbers the problems of its last section 1 through 6 rather than 43 through 51, and its Table 2 runs to order eight rather than seven. The numbering below is the journal numbering, so the version is load bearing. The eight authors and the title match both records.

Definitions

Definition 4 reads:

A permutation of the multiset {1, 1, 2, 2, 3, 3, . . . , n, n} is a Stirling permutation of order n if every value j appearing between the two instances of i satisfies j > i.

Q_n denotes the set of Stirling permutations of order n, written as a word w(1) w(2) ⋯ w(2n). Such a word is read as a parking preference list on 2n spots: car i drives to spot w(i) and takes the first free spot at or after it. Car i is lucky when it parks at w(i), and d(i) is the spot it takes minus w(i). Definition 39 sets dis(w) = (d(1), …, d(2n)), the displacement composition. So d(i) = 0 exactly when car i is lucky, and the number of nonzero entries of dis(w) is the number of unlucky cars.

The statement in question

Corollary 41 reads:

For every w ∈ Q_n, we have 1 ≤ |{i ∈ [2n] : d(i) = 0}| ≤ n.

Equivalently, the number of nonzero entries of dis(w) lies between n and 2n - 1. The article then says “The bound determined in Corollary 41, suggests the following avenue of research” and states:

Problem 48. Determine if, for i ∈ [n, 2n − 1], there exists w ∈ Q_n such that dis(w) has exactly i nonzero entries. Equivalently, can we always find a Stirling permutation with i unlucky cars?

The text immediately after Problem 48 points to Table 3 for the case n = 3, and Problem 49 is posed as its follow-up: for k ∈ [2, n - 1], count the w ∈ Q_n whose displacement composition has k zero parts. Problem 49 asks for the sizes of the fibres; Problem 48 asks only whether they are all nonempty.

What the article already settles at the two ends

Theorem 13 gives (n - 1)! Stirling permutations whose displacement composition has exactly one zero entry, and Corollary 45 gives C_n of them with exactly n nonzero parts. So the two extreme counts 2n - 1 and n are attained; Problem 48 is about the counts strictly between them.

Two further statements of the article constrain any answer. The displacement of any Stirling permutation of order n sums to n^2, and the first entry of dis(w) is always zero.

Scope of the recorded answer

Every count in [n, 2n - 1] is attained. For j from 0 to n - 1 take

V(n, j) = (j+1)(j+1) (j+2)(j+2) ⋯ n n  j j (j-1)(j-1) ⋯ 1 1 ,

each value written twice in succession, the values above j ascending and the values at most j descending. Equal letters are adjacent, so no value stands between two equal letters and V(n, j) ∈ Q_n. Parking it leaves j + 1 lucky cars, hence 2n - j - 1 nonzero entries, and j ↦ 2n - j - 1 carries [0, n - 1] onto [n, 2n - 1].

At n = 3 the three words are 112233, 223311 and 332211, with displacement compositions (0,1,1,2,2,3), (0,1,1,2,0,5) and (0,1,0,3,0,5); each sums to 9 = 3^2, as the article’s displacement identity requires, and they carry five, four and three nonzero entries.

Bounded prior-resolution evidence

The journal article and the sole arXiv version were opened, together with arXiv:2410.08057 (parking functions with a fixed set of lucky cars, which does not treat Stirling permutations), arXiv:2507.17667 (symmetric decompositions and Euler–Stirling statistics, which treats neither parking nor luck) and arXiv:2508.13917. None records an answer to Problem 48, and no later version of the article exists. Citation-index result pages were not reachable, so this is a bounded negative finding and no worldwide priority claim is made.