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Partial Deranged Bell Numbers and Their Combinatorial Properties

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Pith's one-line read The paper's central claim is that partial deranged Bell numbers unify derangements, Stirling numbers, ordered Bell numbers, and complementary Bell numbers through a two-term difference identity.

desk verdict A natural new sequence with a correct EGF, but the headline identity and several theorems are false as written due to binomial/Stirling mix-ups and a sign error. read the letter →

arxiv 2507.21643 v1 pith:LU36NWYY submitted 2025-07-29 math.CO

Yahia Djemmada , Levent Karg{i}n , Mümün Can This is my paper
classification math.CO MSC 05A1805A1911B7311B7511B68
keywords partialderangedBellnumberssetpartitionsderangementsStirlingofthesecondkindorderedcomplementaryMellinderivativeWilfconjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces partial deranged Bell numbers, which count set partitions of $[n]$ in which exactly $r$ of the blocks stay put when the blocks are permuted while the remaining blocks are all moved. The central claim is that these numbers unify partial derangements, Stirling numbers of the second kind, deranged Bell numbers, and ordered Bell numbers, with explicit formulas, generating functions, and recurrences. In particular, the paper derives the identity $\tilde{\phi}_n=\tilde{w}_{n,0}-\tilde{w}_{n,1}=\tilde{w}_{n-1,0}-2\tilde{w}_{n-1,2}$, linking the family to complementary Bell numbers and to Wilf's conjecture. A polynomial analogue connects the numbers to exponential polynomials, geometric polynomials, and Bernoulli numbers, and yields closed forms for finite sums involving Stirling numbers and binomial coefficients. The family thus gives a common combinatorial home for several classical sequences.

What carries the argument

The main analytic tool is the Mellin derivative operator $(xD)^n f(x)=x\frac{d}{dx}f(x)$, applied to the generating function $f(x)=\frac{e}{r!}(x-1)^r e^{-x}/(2-x)$ and evaluated at $x=1$. The operator turns the exponential generating function into a formula for $\tilde{w}_{n,r}$, and its Leibniz rule combines with evaluations of $(xD)^n e^{-x}$ and $(xD)^n(2-x)^{-1}$ to produce the complementary-Bell identities. In the polynomial section the machinery is expanded to r-exponential polynomials $\phi_{n,r}(x)=\sum_k \left\{n+r\atop k+r\right\}_r x^k$, geometric polynomials $w_n(x)=\sum_k \left\{n\atop k\right\}k!x^k$, and higher-order Bernoulli numbers, which jointly generate the identities and finite-sum formulas.

What would settle it

Compute $\tilde{w}_{2,1}$ directly from the definition: only the one-block partition contributes, so $\tilde{w}_{2,1}=1$, while the combinatorial count in Theorem 3 gives $\binom{2}{1}\binom{1}{1}\tilde{w}_{1}+\binom{2}{2}\binom{2}{1}\tilde{w}_{0}=0+2=2$. The printed binomial-based recurrence therefore fails at $n=2, r=1$.

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Extended reading notes

Core claim

The paper's discovery is a two-parameter sequence $\tilde{w}_{n,r}$ whose second parameter records how many blocks of an ordered partition remain fixed. Formally, $\tilde{w}_{n,r}=\sum_{k=0}^{n}\left\{n\atop k\right\}d_{k,r}$, where $\left\{n\atop k\right\}$ is a Stirling number of the second kind and $d_{k,r}$ is the number of permutations of $k$ objects with exactly $r$ fixed points. The paper proves an exponential generating function, $\sum_{n\ge0}\tilde{w}_{n,r}t^n/n!=(e^t-1)^r e^{-(e^t-1)}/(r!(2-e^t))$, and from it derives recurrences and identities relating the sequence to deranged Bell numbers, ordered Bell numbers, complementary Bell numbers, r-exponential polynomials, geometric polynomials, and Bernoulli numbers. The central result is the two-term difference identity $\tilde{\phi}_n=\tilde{w}_{n,0}-\tilde{w}_{n,1}=\tilde{w}_{n-1,0}-2\tilde{w}_{n-1,2}$.

Load-bearing premise

The derivations assume that the chosen elements that will form the r fixed blocks can be divided among those blocks in $\binom{k}{r}$ ways, when the actual number of ways to split k labelled elements into r nonempty unlabelled blocks is a Stirling number of the second kind.

Editorial extensions

If this is right

  • The family interpolates between deranged Bell numbers at $r=0$ and ordered Bell numbers; summing $\tilde{w}_{n,r}$ over $r$ gives the ordered Bell number $w_n$, so every ordered partition splits uniquely by its number of fixed blocks.
  • The identity $\tilde{\phi}_n=\tilde{w}_{n,0}-\tilde{w}_{n,1}$ expresses complementary Bell numbers as a difference of two partial deranged Bell numbers, giving Wilf's conjecture a formulation in terms of the new family.
  • The polynomial identity $\sum_{r=0}^{n} r\,\tilde{w}_{n,r}(y)=w_n(y)$ shows that the total number of fixed blocks over all ordered partitions of $[n]$ equals the ordered Bell number, a direct corollary of the generating function.
  • The finite-sum formulas in Corollaries 14 and 21 evaluate sums of products of binomial coefficients and Stirling or Bernoulli numbers in terms of partial derangement numbers, so they can be used to simplify or verify other combinatorial sums.
  • The exponential generating function gives a route to asymptotic or probabilistic information about the distribution of fixed blocks among ordered partitions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I would expect the combinatorial proofs to require the Stirling number $S(k,r)$ rather than the binomial coefficient $\binom{k}{r}$ when counting the possible fixed blocks; replacing it would preserve the analytic generating-function results while changing the recurrences in Theorems 3 and 6.
  • The difference identity suggests a concrete attack on Wilf's conjecture: study the sign and size of $\tilde{w}_{n,0}-\tilde{w}_{n,1}$ through the polynomial recurrences, since the conjecture is exactly that this difference never vanishes for $n\neq2$.
  • The geometric-polynomial integral identity $\sum_r (-1)^r d_r \tilde{w}_{n,r}(y)=(w_n(y)+w_n(-y))/2$ could be read as an expectation over a Poisson process, giving a probabilistic interpretation of the partial deranged Bell numbers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper introduces partial deranged Bell numbers \tilde{w}_{n,r}, defined as the number of set partitions of [n] whose blocks are permuted with exactly r fixed blocks, and studies their generating functions, recurrences, and polynomial analogues. The main advertised results are a formula \tilde{w}_{n,r} = \sum_k \binom{n}{k}\binom{k}{r}\tilde{w}_{n-k}, an identity relating \tilde{w}_{n,r} to complementary Bell numbers, and the headline identity \tilde{\phi}_n = \tilde{w}_{n,0}-\tilde{w}_{n,1} = \tilde{w}_{n-1,0}-2\tilde{w}_{n-1,2}. The object and the exponential generating function in (10) are correct, and Proposition 2 is a valid double-counting argument. However, several central theorems are false as stated because the paper repeatedly replaces the Stirling number S(k,r) by the binomial coefficient \binom{k}{r}, and the headline identity has a sign error. These errors invalidate the main results of the paper in its current form.

Significance. If the combinatorial framework were repaired, the paper could provide a modest extension of deranged Bell numbers with some useful identities connecting partitions, derangements, and ordered Bell numbers. The intended corrections (replacing \binom{k}{r} by S(k,r) in several displayed formulas and correcting the sign in the complementary-Bell identity) seem straightforward, and the underlying object is coherent. However, the manuscript as submitted proves false statements in its abstract, in Theorem 3, in Theorem 6, in Theorem 10, and in Theorem 12, so the current version cannot be considered a sound contribution without substantial revision.

major comments (4)
  1. [Theorem 3] Theorem 3 is false as stated. For n=2, r=1, the left-hand side is \tilde{w}_{2,1}=1, while the right-hand side is \binom{2}{1}\binom{1}{1}\tilde{w}_1 + \binom{2}{2}\binom{2}{1}\tilde{w}_0 = 2. The proof says that partitioning k chosen elements into r fixed blocks can be done in \binom{k}{r} ways; the correct count is the Stirling number S(k,r). With S(k,r) in place of \binom{k}{r}, the identity can be repaired, but the printed formula and proof are wrong.
  2. [Equation (8) and Theorem 6] Equation (8) is false: the correct Mellin-derivative evaluation is (xD)^n((x-1)^k)|_{x=1} = k! S(n,k), not k!\binom{n}{k}. For example, n=2,k=1 gives left-hand side 1 but right-hand side 2. This error propagates into Theorem 6: equation (12) with \binom{k}{r} is false, since for n=2,r=1 the left-hand side is \tilde{w}_{2,1}-2\tilde{w}_{2,2}=-1 while the right-hand side is 0. The derivation should produce a summation with S(j,r), not \binom{j}{r}.
  3. [Abstract and Remark 7] The headline identity \tilde{\phi}_n = \tilde{w}_{n,0}-\tilde{w}_{n,1} = \tilde{w}_{n-1,0}-2\tilde{w}_{n-1,2} is false for n=3: \tilde{\phi}_3=1, whereas \tilde{w}_{2,0}-2\tilde{w}_{2,2}=1-2=-1. A direct EGF computation gives \tilde{w}_{n,1}-2\tilde{w}_{n,2} = -\tilde{\phi}_n - \tilde{\phi}_{n+1} and hence \tilde{w}_{n-1,0}-2\tilde{w}_{n-1,2}=-\tilde{\phi}_n. This is not a minor typo: the central claimed connection to complementary Bell numbers has the wrong sign and the wrong index.
  4. [Theorems 10 and 12] The same binomial/Stirling confusion appears in the polynomial section. Theorem 10 is false: for n=2, m=0, r=1, the left-hand side is \tilde{w}_{2,1}(y)=y, while the right-hand side is 2y. The correct recurrence has a factor 1/\binom{m+r}{r} and the Stirling number S(k,r) in the sum. Theorem 12 is also false: for n=2,r=1, the left-hand side equals -1 while the right-hand side, with \binom{k}{r}, equals 0. Corollary 14 inherits these errors. These are load-bearing failures, not presentation issues.
minor comments (4)
  1. [Lemma 5] The statement should require n \ge 0 rather than 'all integers n', since Mellin derivatives are applied to functions at x=1 and the coefficients are nonnegative integers.
  2. [Theorem 4 proof] The inclusion-exclusion proof writes A_i as the set of partitions where the i-th block is fixed, but the number of blocks k varies with the partition; the notation should make the dependence on k explicit.
  3. [Proposition 15 proof] The sentence 'Setting z \to 1-z in (26)' is confusing because z is a formal variable; the variable substitution should be described more carefully before integrating with respect to z.
  4. [Throughout] There are frequent typographical inconsistencies between \tilde{w}_{n,r}, \Tilde{w}_{n,r}, and other tilde notations, which make some displays hard to read.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the partial deranged Bell numbers and their identities are derived from explicit definitions and standard external identities, not from fitted parameters or self-referential uniqueness arguments.

full rationale

The paper defines w~_{n,r} = sum_{k=0}^n S(n,k)d_{k,r} in Proposition 2 and derives the exponential generating function (10) directly from that definition plus the standard EGF for Stirling numbers and the EGF (1) for partial derangements. Lemma 5 and Theorems 6, 8, 9, 10, 12, 15, 18, and 20 are subsequent Mellin-operator, coefficient-comparison, or convolution manipulations of (10) together with cited external identities (e.g., equations (4), (9), (11), (14), (16), (18)). The complementary Bell numbers are defined independently by sum (-1)^k S(n,k), so the claimed identity w~_{n,0}-w~_{n,1}=phi~_n is a derived theorem whose two sides are not equal by definition. The only self-citation is the prior deranged Bell numbers [4], used as a baseline sequence whose definition is restated inside Proposition 2; it is a building block, not a justification of the new conclusions. There are no fitted parameters, no data subsets, and no uniqueness theorem imported from the authors. I note, for correctness rather than circularity, that some displayed identities are arithmetically false as printed (e.g., the binomial count in Theorem 3 and the sign in the central identity), but a false derivation is not a circular one. The derivation chain is self-contained with respect to the stated definitions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No free parameters or fitted constants appear; the results are pure identities. The axioms are standard generating-function and Mellin-derivative tools, plus one domain convention about labeling blocks. The main load-bearing weakness is not an axiom but an incorrect count in the proofs.

assumptions (5)
  • standard math Exponential generating function for Stirling numbers of the second kind: sum_{n>=k} S(n,k) t^n/n! = (e^t-1)^k/k!.
    Used to derive EGF (10) in Lemma 5 and repeatedly in coefficient comparisons.
  • standard math Mellin derivative identity (xD)^n f(x) = sum_k S(n,k) x^k f^(k)(x), Eq. (4).
    Core tool in Lemma 5 and Theorems 6, 8, 9; cited from Gould [15].
  • standard math Leibniz rule for the Mellin derivative, Eq. (11).
    Used to split products in Lemma 5 and Theorem 8; cited from Boyadzhiev [6].
  • standard math Exponential generating function for Bernoulli numbers, Eq. (29).
    Used in Theorem 20 to derive the Bernoulli connection.
  • domain assumption Combinatorial model where the k blocks of a set partition are treated as labeled positions for the purpose of counting fixed blocks.
    Definition 1 and Proposition 2 require a canonical labeling (e.g., increasing smallest elements) for 'fixed block' to be well-defined.
invented entities (1)
  • Partial deranged Bell numbers \tilde{w}_{n,r}
    purpose: Count set partitions of [n] with exactly r fixed blocks and the remaining blocks deranged; the paper's primary object.
    Definition 1 introduces the sequence as a new object. It is a purely combinatorial definition with no external falsifiable handle; its value is in the identities it satisfies, several of which are misstated in the preprint.

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Pith. "Pith review of Partial Deranged Bell Numbers and Their Combinatorial Properties." pith.science (2026). https://pith.science/paper/LU36NWYY

@misc{pith2026250721643,
  author       = {Pith},
  title        = {Pith review of: Partial Deranged Bell Numbers and Their Combinatorial Properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LU36NWYY}},
  note         = {Machine review of arXiv:2507.21643}
}
abstract

We introduce a novel generalization of deranged Bell numbers by defining the partial deranged Bell numbers $w_{n,r}$, which count the number of set partitions of $\left[ n\right] $ with exactly $r$ fixed blocks, while the remaining blocks are deranged. This construction provides a unified framework that connects partial derangements, Stirling numbers, and ordered Bell numbers. We investigate their combinatorial properties, including explicit formulas, generating functions, and recurrence relations. Moreover, we demonstrate that these numbers are expressible in terms of classical sequences such as deranged Bell numbers and ordered Bell numbers, and reveal their relationship to complementary Bell numbers, offering insights relevant to Wilf's conjecture. Notably, we derive the identity \[ \tilde{\phi}_{n}=\Tilde{w}_{n,0}-\Tilde{w}_{n,1}=\tilde{w}_{n-1,0}-2\tilde {w}_{n-1,2}, \] which illustrates their structural connection to complementary Bell numbers. We also introduce a polynomial expansion for these numbers and explore their connections with exponential polynomials, geometric polynomials, and Bernoulli numbers. These relationships facilitate the derivation of closed-form expressions for certain finite summations involving Stirling numbers of the second kind, Bernoulli numbers, and binomial coefficients, articulated through partial derangement numbers.

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Forward citations

Cited by 2 Pith papers

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    Deranged unit-interval parking functions are the preimage of deranged ordered set partitions; their count equals the deranged Bell numbers, with new leader/lucky-car tests, fixed-block Poisson law, and related refinements.

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