Double Perfect Partitions of Higher Order

#A109 INTEGERS 25 (2025)
DOUBLE PERFECT PARTITIONS OF HIGHER ORDER
Augus ine O. Munagi
School o Ma hema ics, Uni e si y o he Wi wa e s and, Johannesbu g, Sou h
A ica
[email p o ec ed]
Recei ed: 1/9/25, Accep ed: 11/9/25, Published: 11/25/25
Abs ac
A pa i ion o a posi i e in ege nis called double-pe ec i he summands con ain
wo pa i ions o e e y in ege be ween 2 and n−2. In his pape we gi e new
de i a ions o known esul s on double-pe ec pa i ions. Then we conside gene -
alized double-pe ec pa i ions o nin which he summands con ain wo pa i ions
o e e y in ege be ween and n− , whe e 2 ≤ < n/2. Ou esul s include explici
cha ac e iza ions o double-pe ec pa i ions o all o de s and a seemingly new class
o pseudo-pe ec pa i ions ha p oduce double-pe ec pa i ions. We also s a e
an inclusi e enume a ion o mula in e ms o o de ed ac o iza ion unc ions.
1. In oduc ion
Apa i ion o a posi i e in ege nis any nondec easing sequence o posi i e in ege s
whose sum is n. The summands a e called pa s, and nis he weigh , o he pa i ion.
Thus, a pa i ion λo n(also exp essed as λ⊢n) in o kpa s will be deno ed by
λ= (λ1, λ2, . . . , λk),0< λ1≤···≤λk
o
λ= (λm1
1, λm2
2, . . . , λm
),0< λ1<· · · < λ ,1≤ ≤k,
whe e mideno es he mul iplici y o λi o all i.
The de ini ion o a pe ec pa i ion i s appea ed in he wo ks o P. A. MacMa-
hon [5, 6]. Subsequen ly o he ma hema icians s udied and ound se e al p ope ies
and gene aliza ions o pe ec pa i ions (see o example, [1, 2, 4, 7, 8]).
De ini ion 1. Ape ec pa i ion o nis a pa i ion in which he pa s con ain
exac ly one pa i ion o e e y posi i e in ege less han o equal o n.
DOI: 10.5281/zenodo.17711671
INTEGERS: 25 (2025) 2
Fo example, (13,4) ⊢7 is a pe ec pa i ion since i con ains he pa i ions (1),
(12), (13), (4),(1,4),(12,4),(13,4) wi h weigh s 1,2,...,7, espec i ely.
The e is a known bijec ion be ween he se o pe ec pa i ions o nand he
se o o de ed ac o iza ions o N=n+ 1, ha is, ep esen a ions o Nas o de ed
p oduc s o posi i e in ege s wi hou uni ac o s [3, 6, 9]. Fo example, N= 12 has
eigh o de ed ac o iza ions, namely, 12,2·6,6·2,3·4,4·3,2·2·3,2·3·2,3·2·2. Le
n+ 1 = a1a2···a , ai>1, be an o de ed ac o iza ion o n+ 1. Then he bijec ion
is gi en by
a1a2···a −→ 1a1−1, aa2−1
1,(a1a2)a3−1,...,(a1a2···a −1)a −1.(1)
Le (n, k) be he numbe o o de ed ac o iza ions o nin o k ac o s, and le
he p ime-powe ac o iza ion o nbe n=pα1
1pα2
2···pα
. The o mula o (n, k)
is gi en by (see [6] o [3, p. 59])
(n, k) =
k−1
X
i=0
(−1)ik
i
Y
j=1 αj+k−i−1
αj.
We de ine (n) := Pk (n, k), whe e (0) = 0 and (1) = 1. So he o mula o he
numbe pe (n) o pe ec pa i ions o nis gi en by
pe (n) = (n+ 1).
Example 1. Table 1 shows he o de ed ac o iza ions o 6 which co espond o he
pe ec pa i ions o 5.
O de ed Fac o iza ion o 6 6 2 ·3 3 ·2
Pe ec Pa i ion o 5 (15) (1,22) (12,3)
Table 1: Fac o iza ions o 6 and pe ec pa i ions o 5
Pa k [8] gene alized pe ec pa i ions o “comple e pa i ions” by emo ing he
uniqueness condi ion om subpa i ions, ha is, con ained pa i ions.
De ini ion 2 (Pa k).A comple e pa i ion o nis a weakly inc easing pa i ion λ
wi h λ1= 1, such ha each in ege m, 1≤m≤n, can be exp essed as a sum o
pa s o λ, ha is, each mcan be exp essed as Pk
j=1 αjλj, whe e αj∈ {0,1}.
Fo example, o he 7 pa i ions o n= 5, ou a e comple e pa i ions, namely,
(15), (13,2), (12,3), (1,22).
Ano he ex ension o pe ec pa i ions was in oduced by Lee [4] based on he
ollowing obse a ion.
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Lemma 1 (Lee).Le H(n, )be he se o pa i ions o n ha con ain exac ly
pa i ions o m, ≤m≤n− , and exac ly one pa i ion o e e y o he posi i e
in ege less han n. Then H(n, )=∅i and only i = 1 o = 2.
The case = 1 gi es pe ec pa i ions. Na u ally, Lee decided o s udy he
seemingly o e looked case o = 2.
De ini ion 3. Adouble-pe ec pa i ion is a pa i ion (λ1, λ2, . . . , λk)⊢nsuch ha
each in ege m, 2≤m≤n−2, can be ep esen ed exac ly wice as m=Pk
i=1 αiλi,
whe e αi∈ {0,1}.
Fo example, (15,2) is a double-pe ec pa i ion o 7 because i con ains wo pa -
i ions o 2,3,4,5 and one pa i ion o 1,6,7:
(1),(12),(2),(13),(1,2),(14),(12,2),(15),(13,2),(14,2),(15,2).(2)
P oposi ion 1 (Lee [4]).A double-pe ec pa i ion has he o m
(1q1,2q2,(q1+2q2−1)q3,{(q1+2q2−1)(q3+1)}q4,{(q1+2q2−1)(q3+1)(q4+1)}q5, . . .),
(3)
whe e q1≥2and q2, q3, . . . a e posi i e in ege s such ha q1= 3 implies q2= 1.
Theo em 1 (Lee [4]).Le d(n)be he numbe o double-pe ec pa i ions o a
posi i e in ege n. We ha e
d(n) = ( (n−1) i n≡ 1 (mod 4),
(n−1) − (n−1
4)i n≡1 (mod 4).(4)
In he cou se o p o ing Theo em 1, Lee sepa a ed (3) in o wo o ms o double-
pe ec pa i ions λby se ing q2>1 (wi h q1= 3) and q2= 1, as ollows:
λ= (13,2q2,(2(q2+ 1))q3,(2(q2+ 1)(q3+ 1))q4,...,(2(q2+ 1) ···(qk−1+ 1))qk);
(5)
λ= (1q1,2,(q1+ 1)q3,((q1+ 1)(q3+ 1))q4,...,((q1+ 1)(q3+ 1) ···(qk−1+ 1))qk).
(6)
These o ms we e hen shown o co espond o he ollowing o de ed ac o iza ions:
n−1 = 2(q2+ 1)(q3+ 1) ···(qk−1+ 1)(qk+ 1), q2>1; (7)
n−1=(q1+ 1)(q3+ 1)(q4+ 1) ···(qk−1+ 1)(qk+ 1), q1>1.(8)
No ice ha (7) and (8) exclude ac o iza ions o he ype n−1=2·2·(q3+ 1) ···.
The numbe o such ac o iza ions, (n−1
2), is he e o e sub ac ed om he o al
coun in (4).
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The aim o his pape is o s udy gene alized double-pe ec pa i ions o n ha
con ain wo pa i ions o e e y in ege be ween + 1 and n− −1, whe e 1 ≤ ≤
⌊n−2
2⌋. These will be called double-pe ec pa i ions o o de .
We will i s gi e new p oo s o Theo em 1 and P oposi ion 1 in Sec ion 2. Then
in Sec ion 3 we adap he new app oach o he s udy o double-pe ec pa i ions
o o de and cha ac e ize he i s sub-class o he pa i ions ollowed by an enu-
me a ion esul (Theo ems 2 and 3). In Sec ion 4 we discuss al e na i e me hods o
gene a ing he pa i ions (Theo em 4). Sec ion 5 deals wi h a special o de ed ac-
o iza ion which leads o he second sub-class o double-pe ec pa i ions o o de
. Finally, we s a e an inclusi e enume a ion o mula (Theo em 7).
2. New P oo s o Lee’s Resul s
Le G(λ) be he se o nonemp y subpa i ions o λ⊢n. Thus, i λis comple e,
hen G(λ) con ains a leas one pa i ion o e e y posi i e in ege less han o equal
o n.
We will show ha double-pe ec pa i ions o Na ise om pe ec pa i ions o
N−2, and hence om o de ed ac o iza ions o N−1 by (1). Le Pe (n) deno e he
se o pe ec pa i ions o n, and le D(N) be he se o double-pe ec pa i ions
o N.
P oposi ion 2. A double-pe ec pa i ion λ⊢N > 3may be ob ained om a
pa i ion β∈Pe (N−2) in wo ways:
I. I he mul iplici y o 1 in βis 1, hen inse 12in o β. Deno e he esul ing
se by E(12).
II. I βdoes no con ain 2 as a pa , inse 2in o β. Deno e he esul ing se by
W(2).
Then
D(N) = E(12)∪W(2).
P oo . Le h(m) be he pa i ion o mcon ained in G(β) and w i e λ∪γ o he
pa i ion ob ained by combining he pa s o wo pa i ions λand γ. Assume ha
λ⊢Nis ob ained om β∈Pe (N−2) by inse ion o 12o 2 acco ding o I o II
espec i ely.
Then om G(β) o G(λ) we ind one addi ional pa i ion o each j∈ {2,3, . . . , N−
2}, namely (12)∪h(j−2) o (2)∪h(j−2). Then one new pa i ion o each o N−1
and Nappea s, ha is, (12)∪h(N−3) o (2) ∪h(N−3) and (12)∪h(N−2) o
(2) ∪h(N−2).
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So he esul ing pa i ion λis double-pe ec by de ini ion. Fo example, le
β= (12,3). Then om II, λ= (2) ∪β= (12,2,3), and ou cons uc ion is shown in
Table 2.
j h(j)∈G(β)γ∈G(λ) G(β)
1 (1) −
2 (12) ((2), h(0)) = (2)
3 (3) ((2), h(1)) = (1,2)
4 (1,3) ((2), h(2)) = (12,2)
5 (12,3) ((2), h(3)) = (2,3)
6−((2), h(4)) = (1,2,3)
7−((2), h(5)) = (12,2,3).
Table 2: The cons uc ion in he p oo o P oposi ion 2 o β= (12,3)
Example 2. We illus a e P oposi ion 2 u he by ex ending Table 1 o he co -
esponding double-pe ec pa i ions (see Table 3).
O de ed Fac o iza ion o 6 6 2 ·3 3 ·2
Pe (5) (15) (1,22) (12,3)
Inse Pa s 2 122
Double-Pe ec Pa i ion o 7 (15,2) (13,22) (12,2,3)
Table 3: Fac o iza ions o 6 and double-pe ec pa i ions o 7
The ollowing co olla y is equi alen o P oposi ion 1.
Co olla y 1. A double-pe ec pa i ion has one o he o ms in (5) and (6).
P oo . Conside an o de ed ac o iza ion o he o m N−1=2a2a3···ak, a2>2.
F om (1) he co esponding pe ec pa i ion is
(1,2a2−1,(2a2)a3−1,(2a2a3)a4−1,...,(2a2a3···ak−1)ak−1).(9)
Secondly, an o de ed ac o iza ion o he o m N−1=a1a2a3···ak, a1>2, co e-
sponds o he pe ec pa i ion
(1a1−1, aa2−1
1,(a1a2)a3−1,...,(a1a2a3···ak−1)ak−1).(10)
Obse e ha he pa i ions in (9) and (10) ul ill he asse ed p ope ies o βin
pa s I and II o P oposi ion 2. Las ly, inse ion o 12and 2 in o hese pa i ions
es o es (5) and (6) espec i ely (on se ing ai=qi+ 1 o all i).

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P oo o Theo em 1. P oposi ion 2 implies ha d(N)= (N−1) wi h he excep ion
o ce ain duplica ed pa i ions. No e ha he ac o iza ions N−1 = 2 ·2·mand
N−1=4·mp oduce he same double-pe ec pa i ion:
N−1=2·2·m=⇒β= (1,2,4m−1)7−→ (13,2,4m−1)∈E(12)
and
N−1=4·m=⇒β= (13,4m−1)7−→ (13,2,4m−1)∈W(2).
The numbe o ac o iza ions o he o m N−1 = 2·2·mis gi en by (N−1
4). Thus,
when N−1≡0 (mod 4), we ha e d(N) = (N−1) − (N−1
4). This comple es he
p oo .
3. Double Pe ec Pa i ions o Highe O de
We p opose he ollowing ex ension o double-pe ec pa i ions.
De ini ion 4. Adouble-pe ec pa i ion o o de is a pa i ion λ= (λ1, . . . , λk)⊢
Nsuch ha each in ege mwi h + 1 ≤m≤N− −1 can be ep esen ed exac ly
wice as m=Pk
i=1 αiλi, αi∈ {0,1}, and o he in ege s less han o equal o Ncan
be uniquely ep esen ed.
In pa icula , double-pe ec pa i ions o o de = 1 a e he o iginal double-pe ec
pa i ions discussed abo e.
The ep esen a ion scheme o a double-pe ec pa i ion o No o de is
1,2, . . . ,
| {z }
1 ime
, + 1, + 2, . . . , N − −1
| {z }
2 imes
, N − , N − + 1,...,N −1, N
| {z }
1 ime
.(11)
Le U (N) be he se o double-pe ec pa i ions o No o de wi h u (N) =
|U (N)|. Then (11) implies ha
U (N)=∅⇐⇒N≥2( + 1).(12)
Le D (N) be he subse o U (N) con aining pa i ions which may be ound by in-
se ions o (1 +1) and ( +1) in o pe ec pa i ions ( hus ex ending he cons uc ion
in Sec ion 2 ha co esponds o = 1). Then de ine E (N) := U (N) D (N).
Pa i ions in D (N) and E (N) will also be e e ed o as Type-A and Type-
B espec i ely. The es o his sec ion is de o ed o he cha ac e iza ion and
enume a ion o D (N). P ope ies o E (N) will be explo ed in de ail in Sec ions
4 and 5.
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Theo em 2. A Type-A double-pe ec pa i ion λ⊢No o de > 0may be
ob ained om a pa i ion β∈Pe (N− −1) in wo ways:
I. I he mul iplici y o 1 in βis , hen inse 1 +1 in o β, and deno e he esul ing
se by A(1 +1).
II. I he mul iplici y o 1 in βis di e en om , hen inse + 1 in o β, and
deno e he esul ing se by B( + 1).
Then
D (N) = A(1 +1)∪B( + 1).(13)
P oo . Le h(m)∈G(β),1≤m≤N− −1. We show ha any λ⊢Nob ained
om I o II is double-pe ec o o de by accoun ing o new pa i ions a ising
be ween G(β) and G(λ).
The single pa i ions o 1,2, . . . , a e no a ec ed by inse ion o addi ional pa s
in o β, bu one new pa i ion o each m∈ { + 1, + 2, . . . , N − −1}appea s
om A(1 +1) o B( + 1), namely, (1 +1) and (1 +1)∪h(m− −1) o ( + 1) and
( +1)∪h(m− −1)), espec i ely.
Finally we ob ain one new pa i ion o each m∈ {N− , . . . , N}by symme y
(since G(λ) al eady con ains pa i ions o j= 0,1,..., ). This shows ha weigh s
o pa i ions in G(λ) a e dis ibu ed as in (11), as desi ed.
Rema k 1. In he p oo o Theo em 2 conside he e ec o inse ing γ∈P( +
1) {(1 +1),( + 1)}in o β, whe e > 1. So γhas he o m γ= (γ1, . . . , γk), k > 1
and γi>1 o some i.
We claim ha λ=γ∪βis no a Type-A double-pe ec pa i ion o o de .
Assume ha he mul iplici y o 1 in βis x≥ . Then λwould con ain a leas
wo pa i ions o γiins ead o one, namely (1γi) and (γi).
The case when he mul iplici y o 1 in βis x < a ec s only ype II. Obse e
ha al eady x+ 1 ∈βsince βis pe ec . Thus, i 1 ∈γ, hen λwould con ain a
leas wo pa i ions o x+ 1: (1x+1) and (x+ 1).
Howe e , i 1 /∈γwhen x< , i is possible o λ o be double pe ec o o de ,
bu no o Type-A. Fo example, conside N= 23, = 5 wi h β= (12,3,62) and
γ= (32). Then i may be e i ied ha λ= (12,33,62)∈E5(23).
A sys ema ic me hod o ob aining all membe s o E (N) is discussed in Sec ion 5.
Co olla y 2. A Type-A double-pe ec pa i ion λ∈D (N)has ei he o he ol-
lowing o ms:
λ= (12 +1,( + 1)a2−1,(( + 1)a2)a3−1,...,(( + 1)a2a3···ak−1)ak−1),(14)
λ= (1a1−1,( + 1), aa2−1
1,(a1a2)a3−1,(a1a2a3)a4−1,...,(a1a2···ak−1)ak−1),(15)
whe e a1= + 1 and he loca ion o + 1 in (15) depends on i s ela i e size.
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P oo . The wo o ms a e consequences o con e ing he ollowing o de ed ac o -
iza ions o pe ec pa i ions by means o he bijec ion (1), and hen inse ing 1 +1
and + 1 espec i ely.
N− = ( + 1)a2a3···ak; (16)
N− =a1a2a3···ak, a1= + 1.(17)
No e ha he wo se s on he igh -hand-side o (13) a e no always disjoin as
ce ain pa i ions may be ob ained wice using he wo me hods. The ollowing
esul gi es he exac ca dinali y o D (N) a e excluding duplica es.
Theo em 3. The numbe d (N)o Type-A double-pe ec pa i ions o No o de
, 1≤ ≤ ⌊n−2
2⌋, is gi en by
d (N) =





(N− )− (N−
2( +1) )−( ( +1)−1) (N−
+1 )i N≡ (mod 2( + 1)),
(N− )−( ( +1)−1) (N−
+1 )i N≡ −1 (mod 2( + 1)),
(N− )o he wise.
(18)
In pa icula when 1 ≤ ≤3, we ob ain d1(N)=d(N) (same as (4));
d2(N) = ( (N−2) − (N−2
6) i N≡2 (mod 6),
(N−2) o he wise; (19)
d3(N) = 




(N−3) − (N−3
8)− (N−3
4) i N≡3 (mod 8),
(N−3) − (N−3
4) i N≡7 (mod 8),
(N−3) o he wise.
(20)
P oo . F om Theo em 2, λ∈D (N) is ob ained by inse ing 1 +1 o + 1 in o
sui able pe ec pa i ions o n=N− −1. The la e may be cons uc ed om
he o de ed ac o iza ions o N− ; see Co olla y 2.
Thus, d (N)= (N− ) subjec o he ollowing excep ions.
(i) I N− ≡0 (mod 2( + 1)), he ac o iza ions N− = ( +1)·2·mand
N− = (2( + 1)) ·mp oduce he same λ:
( +1)·2·m=⇒(1 , + 1,(2( + 1))m−1)7→ (12 +1, + 1,(2( + 1))m−1)
∈A(1 +1),
and
(2( +1))·m=⇒(12 +1,(2( +1))m−1)7→ (12 +1, +1,(2( +1))m−1)∈B( +1).
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We emo e he i s ype o such ac o iza ions which is coun ed by (N−
2( +1) ).
(ia) Fu he mo e, since N− ≡0 (mod 2( + 1)) implies N− ≡0 (mod
+ 1) we isola e a se o pe ec pa i ions ha do no con ibu e o disco e -
ing addi ional λ, namely, ac o iza ions o he o m a1a2···a m, > 1, whe e
a1a2···a = + 1. No e ha a1a2···a m ansla es in o he pe ec pa i ion
β= (1a1−1, aa2−1
1,(a1a2)a3−1,...,(a1a2···a )m−1). Howe e , βcon ains 1a1−1
bu a1−1= , so Me hod I does no apply. Also, βal eady con ains he pa
+1=a1a2···a , so Me hod II does no apply. The numbe o such non-
con ibu ing pe ec pa i ions o N− −1 is equal o he numbe o o de ed
ac o iza ions o + 1 in o wo o mo e ac o s imes he numbe o o de ed ac o -
iza ions o (N− )/( + 1), ha is, ( ( + 1) −1) (N−
+1 ).
Pa s (i) and (ia) oge he gi e he i s line o he s a ed o mula.
(ii) When N− ≡ + 1 (mod 2( + 1)), we ob ain pa (ia) independen ly. Hence
he second line o he o mula ollows.
The e a e no o he excep ions. The emaining ac o iza ions all yield alid pa -
i ions λ. Hence hei numbe is (N− ).
Example 3. Le N= 17 wi h 1 ≤ ≤7. The membe s o D (N) a e shown
in Table 4. The de i a ion o membe s o D2(17) is shown in Table 5. I may be
e i ied ha he dis ibu ion o weigh s o membe s o G(λ), o e e y λ∈D2(17),
co esponds o he scheme (c . (11))
1,2
|{z}
1 ime
,3,4,...,14
| {z }
2 imes
,15,16,17
| {z }
1 ime
.
D (17) d (17)
1 (13,27),(13,2,43),(13,23,8),(13,2,4,8),(115,2),(17,2,8) 6
2 (114,3),(15,34),(14,3,5) 3
3 (113,4),(1,26,4),(16,4,7) 3
4 (112,5) 1
5 (111,6),(1,25,6),(12,23,6),(13,42,6),(1,2,42,6) 5
6 (110,7) 1
7 (19,8),(1,24,8),(14,5,8) 3
Table 4: Type-A double-pe ec pa i ions o 17 o all o de s
Rema k 2. No e ha D (17) = U (17) when = 2, ha is, E (17) = ∅; bu
E2(17) = {(1,22,34)}. Hence d2(17) = 3 bu u2(17) = 4 (see Example 4).
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[4] H. Lee, Double pe ec pa i ions, Disc e e Ma h. 306 (5) (2006), 519–525.
[5] P. A. MacMahon, Combina o y Analysis, Volume 1, Camb idge Uni e si y P ess, 1915
[6] P. A. MacMahon, The heo y o pe ec pa i ions and he composi ions o mul ipa i e num-
be s, Messenge Ma h. 20 (1891) 103–119.
[7] A. O. Munagi, Pe ec composi ions o numbe s, J. In ege Seq. 23 (2020), A icle 20.5.1.
[8] S. K. Pa k, Comple e pa i ions, Fibonacci Qua . 36 (1998), 354–360.
[9] J. Rio dan, An In oduc ion o Combina o ial Analysis, Wiley, 1958.
[10] OEIS Founda ion Inc. (2025), The On-Line Encyclopedia o In ege Sequences, Published
elec onically a h ps://oeis.o g.