Partial Deranged Bell Numbers

• Partial deranged Bell numbers are two-parameter refinements that count set partitions of [n] with exactly r fixed blocks and deranged remaining blocks.
• They are derived using Stirling transforms, exponential generating functions, and convolution identities to connect fixed-block statistics with classical polynomial families.
• These numbers bridge various combinatorial frameworks by linking deranged partitions, ordered Bell numbers, complementary Bell numbers, and Bernoulli identities.

Partial deranged Bell numbers, denoted $\widetilde{w}_{n,r}$, are a two-parameter refinement of deranged Bell numbers in which one counts set partitions of $[n]$ with exactly $r$ fixed blocks while the remaining blocks are deranged. Introduced as the numbers of $r$-partial deranged partitions, they interpolate between ordinary deranged Bell numbers $\widetilde{w}_{n}=\widetilde{w}_{n,0}$ and ordered Bell numbers obtained by summing over all admissible numbers of fixed blocks. Their theory combines Stirling transforms, exponential generating functions, polynomial refinements, and convolution identities, and it places block-fixed-point statistics on ordered partitions in direct contact with complementary Bell numbers, exponential polynomials, geometric polynomials, and Bernoulli numbers (Djemmada et al., 29 Jul 2025).

1. Definition and combinatorial interpretation

An $r$-partial deranged partition of $[n]=\{1,2,\dots,n\}$ is defined by first partitioning $[n]$ into blocks $B_1,\dots,B_k$, then permuting the blocks, but requiring that exactly $r$ of the blocks stay in their original position under the block permutation. Equivalently, if

$[n]$0

then $[n]$1 for exactly $[n]$2 indices $[n]$3. The number of such objects is $[n]$4, the $[n]$5-partial deranged Bell number, or $[n]$6-PDB number (Djemmada et al., 29 Jul 2025).

This definition generalizes the ordinary deranged Bell numbers. When $[n]$7, no block is fixed, so one recovers the deranged partitions counted by $[n]$8. When $[n]$9 varies, the family records the number of fixed blocks among the ordered blocks of a partition. The relevant permutation statistic is the classical number $r$0 of permutations of $r$1 objects with exactly $r$2 fixed points: $r$3 Accordingly, partial deranged Bell numbers refine the ordered-partition count by replacing the factor $r$4 with the fixed-point statistic $r$5 (Djemmada et al., 29 Jul 2025).

2. Stirling transforms and structural decompositions

The first basic formula is a double-counting identity. If a partition of $r$6 has $r$7 blocks, then there are $r$8 ways to choose the partition and $r$9 ways to permute those $r$0 blocks with exactly $r$1 fixed blocks. Hence

$r$2

This exhibits $r$3 as a Stirling transform of the partial derangement numbers and shows immediately that $r$4 (Djemmada et al., 29 Jul 2025).

A second central identity expresses $r$5 through ordinary deranged Bell numbers: $r$6 Combinatorially, one selects $r$7 elements from $r$8, partitions those $r$9 elements into $\widetilde{w}_{n}=\widetilde{w}_{n,0}$0 fixed blocks in $\widetilde{w}_{n}=\widetilde{w}_{n,0}$1 ways, and lets the remaining $\widetilde{w}_{n}=\widetilde{w}_{n,0}$2 elements form a deranged partition counted by $\widetilde{w}_{n}=\widetilde{w}_{n,0}$3. This makes the ordinary deranged Bell numbers the $\widetilde{w}_{n}=\widetilde{w}_{n,0}$4 base layer of the entire family (Djemmada et al., 29 Jul 2025).

The main specializations can be organized succinctly.

Specialization	Value	Interpretation
$\widetilde{w}_{n}=\widetilde{w}_{n,0}$5	$\widetilde{w}_{n}=\widetilde{w}_{n,0}$6	ordinary deranged Bell numbers
$\widetilde{w}_{n}=\widetilde{w}_{n,0}$7	$\widetilde{w}_{n}=\widetilde{w}_{n,0}$8	ordered Bell numbers
$\widetilde{w}_{n}=\widetilde{w}_{n,0}$9	$r$0	complementary Bell number

These identities show that partial deranged Bell numbers decompose ordered Bell numbers by the number of fixed blocks and recover complementary Bell numbers by finite differences in the $r$1-direction (Djemmada et al., 29 Jul 2025).

3. Generating functions and polynomial refinements

The exponential generating function of $r$2 is

$r$3

At $r$4, this specializes to the generating function for the ordinary deranged Bell numbers: $r$5 The same section also gives an operator representation,

$r$6

which functions as a Mellin-derivative mechanism for the sequence (Djemmada et al., 29 Jul 2025).

To parallel Bell and ordered Bell polynomials, the theory introduces the partial deranged Bell polynomials

$r$7

They satisfy $r$8 and $r$9, the deranged Bell polynomials. Their exponential generating function is

$[n]=\{1,2,\dots,n\}$0

This polynomial extension is the basic analytic tool for the identities in the later parts of the theory (Djemmada et al., 29 Jul 2025).

A bivariate marking of fixed blocks gives

$[n]=\{1,2,\dots,n\}$1

and equivalently

$[n]=\{1,2,\dots,n\}$2

where $[n]=\{1,2,\dots,n\}$3 are the exponential polynomials and $[n]=\{1,2,\dots,n\}$4 are geometric polynomials. These relations encode fixed-block statistics through classical polynomial families (Djemmada et al., 29 Jul 2025).

4. Complementary Bell numbers, ordered Bell numbers, and Bernoulli identities

The ordered Bell numbers are

$[n]=\{1,2,\dots,n\}$5

Partial deranged Bell numbers decompose them by fixed-block count: $[n]=\{1,2,\dots,n\}$6 There is also a weighted decomposition,

$[n]=\{1,2,\dots,n\}$7

and an even/odd symmetrization identity,

$[n]=\{1,2,\dots,n\}$8

Thus ordered Bell numbers are resolved into strata indexed by the number of fixed blocks (Djemmada et al., 29 Jul 2025).

The complementary Bell numbers are

$[n]=\{1,2,\dots,n\}$9

and partial deranged Bell numbers connect to them through

$[n]$0

More generally,

$[n]$1

These formulas are the basis for the paper’s relation to Wilf’s conjecture $[n]$2 for $[n]$3, because they realize complementary Bell numbers as finite differences of the partial deranged Bell hierarchy (Djemmada et al., 29 Jul 2025).

The Bernoulli side enters through the higher-order Bernoulli numbers $[n]$4, defined by

$[n]$5

The paper proves

$[n]$6

and for $[n]$7,

$[n]$8

After substituting the defining expansion of $[n]$9, these become explicit Stirling-number identities involving Bernoulli numbers (Djemmada et al., 29 Jul 2025).

5. Relation to ordinary deranged Bell numbers and combinatorial models

The ordinary deranged Bell numbers are the $B_1,\dots,B_k$0 specialization, but they also have an independent literature. They count ordered partitions

$B_1,\dots,B_k$1

for which $B_1,\dots,B_k$2 is a derangement of $B_1,\dots,B_k$3. Their total number is

$B_1,\dots,B_k$4

and their exponential generating function is

$B_1,\dots,B_k$5

The asymptotic behavior is

$B_1,\dots,B_k$6

with the dominant singularity at $B_1,\dots,B_k$7 (Belbachir et al., 2021).

A later combinatorial development identifies the same ordinary deranged Bell numbers as the counting sequence of deranged unit-interval parking functions. If $B_1,\dots,B_k$8 denotes unit-interval parking functions and $B_1,\dots,B_k$9 ordered set partitions, the explicit bijection

$r$0

restricts to

$r$1

so

$r$2

Here $r$3 is the $r$4-th deranged Bell number in that notation. The same work gives the intrinsic criterion

$r$5

where $r$6 is the leader word, and an equivalent lucky-car criterion

$r$7

It also refines the count by block number and total displacement: $r$8 by singleton blocks: $r$9 and in the $[n]$00-start setting: $[n]$01 These are presented explicitly as refined “partial” deranged Bell numbers in the language of that paper, although the authors mainly formulate them as counts of refined deranged unit-interval parking functions (Djemmada, 30 Jun 2026).

6. Adjacent frameworks, generalizations, and distinctions

Several nearby theories are closely related but not identical to partial deranged Bell numbers. One arithmetic strand comes from congruences between Bell numbers and derangement numbers modulo primes. For every positive integer $[n]$02 and any prime $[n]$03,

$[n]$04

and conversely, for $[n]$05,

$[n]$06

The same work generalizes the result through Touchard polynomials

$[n]$07

with

$[n]$08

That paper does not define partial deranged Bell numbers, but it does isolate truncated Bell and Touchard sums with a clear partial-derangement flavor (Sun et al., 2010).

A second adjacent framework lifts Bell–derangement relations to polynomial congruences via umbral calculus. The bridge identity

$[n]$09

connects Bell polynomials $[n]$10 and derangement polynomials $[n]$11, and the paper derives polynomial analogues of Touchard-type congruences, including

$[n]$12

Again, the terminology “partial deranged Bell numbers” is absent, but the polynomial-and-congruence framework supplies natural ingredients for such a theory (Sun et al., 2010).

A third neighboring direction studies $[n]$13-restricted Bell numbers

$[n]$14

with exponential generating function

$[n]$15

This framework recovers associated, restricted, involution, and derangement-type sequences through appropriate choices of $[n]$16. It is the natural ambient setting for block-size-restricted Bell analogues, but it is distinct from the fixed-block statistic encoded by $[n]$17 (Wakhare, 2017).

A fourth extension introduces higher order degenerate $[n]$18-deranged Bell numbers with singletons and barred preferential arrangements. The master generating function is

$[n]$19

and

$[n]$20

That construction is explicitly distinguished from partial deranged Bell numbers: it is $[n]$21-deranged in the sense of first $[n]$22 singleton blocks and barred-preferential structure, whereas partial deranged Bell numbers count partitions with exactly $[n]$23 fixed blocks (Nkonkobe, 8 May 2026).

Taken together, these neighboring literatures show that the phrase “partial deranged Bell numbers” sits inside a broader Bell–derangement landscape with several non-equivalent parameters: fixed blocks in $[n]$24, no fixed blocks in ordinary deranged Bell numbers, first-$[n]$25 constraints in $[n]$26-deranged Bell numbers, and block-size restrictions in $[n]$27-refined Bell numbers. The specific contribution of $[n]$28 is to make block fixed points themselves the organizing statistic.