Enumerating Parts of \(n\)-Color Partitions

#A94 INTEGERS 25 (2025)
ENUMERATING PARTS OF n-COLOR PARTITIONS
Sh u i Sha ma
Yada ind a Depa men o Sciences, Punjabi Uni e si y Gu u Kashi Campus,
Talwandi Sabo, Punjab, India
sha ma [email p o ec ed]
Amandeep Kau
Depa men o Ma hema ics, Punjabi Uni e si y, Pa iala, Punjab, India
[email p o ec ed]
Recei ed: 2/3/25, Accep ed: 9/25/25, Published: 11/5/25
Abs ac
In ecen yea s, a numbe o au ho s ha e ob ained nume ous ela ions be ween
he numbe o pa s in es ic ed as well as un es ic ed pa i ions and a ious
unc ions om mul iplica i e numbe heo y such as ϕ(m), τ(m), σ(m), e c. In his
pape , we seek simila kinds o ela ions o he numbe o pa s in a ious es ic ed
and un es ic ed n-colo pa i ions. He e, in pa icula , hese ela ions occu wi h
di iso unc ions.
1. In oduc ion
Apa i ion o a posi i e in ege mis a sequence o posi i e in ege s whose sum is m.
While he o de o posi i e in ege s in a pa i ion does no ma e , we con en ionally
lis hem in non-inc easing o de o consis ency. These posi i e in ege s a e also
called he summands o pa s o a pa i ion. Many impo an pa i ion iden i ies
in ol e es ic ions on he pa s o a pa i ion. A e y old and amous iden i y due
o Eule equa es he numbe o pa i ions in o odd pa s o he numbe o pa i ions
in o dis inc pa s. Pa i ion iden i ies also a ise om coun ing he numbe o pa s
in a pa i ion. In he li e a u e, he i s such iden i y is c edi ed o S anley, and
i s gene aliza ion o Elde . To know mo e abou he his o y o hese iden i ies, he
eade is e e ed o [5].
Theo em 1 (S anley’s heo em).The numbe o 1’s in he pa i ions o nequals
he numbe o pa s ha appea a leas once in a gi en pa i ion o n, summed o e
all pa i ions o n.
DOI: 10.5281/zenodo.17535215
INTEGERS: 25 (2025) 2
Theo em 2 (Elde ’s heo em).The numbe o appea ances o a pa kin he
pa i ions o nis equal o he numbe o pa s ha appea a leas k imes in a gi en
pa i ion o n, summed o e all pa i ions o n.
And ews and Me ca’s wo k [3] p o ides a u he gene aliza ion o Elde ’s heo-
em. In [4], he au ho s coun he numbe o e en pa s in pa i ions in o dis inc
pa s. Me ca [6] in es iga ed speci ic es ic ed pa i ions, enume a ing hei pa s
and es ablishing ela ionships be ween hese coun s and he di iso s o m. In [7,8],
he au ho s ob ained nume ous ela ions be ween he numbe o pa s in pa i ions
( es ic ed as well as un es ic ed) and a ious unc ions om mul iplica i e numbe
heo y such as ϕ(m), τ(m), σ(m), e c. In o de o ob ain hese ela ions, hey used
he Lambe se ies, which is gi en by
∞
X
m=1
amqm
1−qm,|q|<1,
whe e he am o m= 1,2,3, . . . a e eal o complex numbe s. This se ies is a
na u al gene aliza ion o he o mula
∞
X
m=1
amqm
1−qm=
∞
X
m=1
b±(m)qm,
whe e
b±(m) = X
d|m
(∓1)1+m/dad.
When wo king wi h n-colo pa i ions, i is na u al o us o seek simila kinds
o ela ions o he numbe o pa s in a ious es ic ed and un es ic ed n-colo
pa i ions. In his manusc ip , howe e , we will ocus only on he di iso unc ions.
Fo a eade who is no exposed o he heo y o n-colo pa i ions, we include
he de ini ion o n-colo pa i ions as gi en in [2]. An n-colo pa i ion o a posi-
i e in ege is a pa i ion, in which a pa o size mcan occu in mdi e en colo s
deno ed by subsc ip s m1, m2, . . . , mm. Fo example, he n-colo pa i ions o 3 a e
31,32,33,22+ 11,21+ 11,11+ 11+ 11.
Le P(m) deno e he numbe o pa i ions o m. Then,
∞
X
m=0
P(m)qm=1
∞
Q
m=1
(1 −qm)
.
As a con en ion, we ake P(0) as 1. Using simila no a ion, we w i e
∞
X
m=0
C(m)qm=1
∞
Q
m=1
(1 −qm)m
,(1)
INTEGERS: 25 (2025) 3
whe e C(m) is he numbe o n-colo pa i ions o mand C(0) = 1.
As o dina y pa i ions ha e been s udied wi h a ious es ic ions, n-colo pa -
i ions oo can be conside ed wi h es ic ions on he pa s and colo s. Gene a ing
unc ions o he numbe o n-colo pa i ions wi h a ious es ic ions we e p o ided
by Aga wal [1]. Howe e , we ake in o accoun some o hese n-colo pa i ions wi h
es ic ions while s udying he numbe o pa s. We include he gene a ing unc-
ions o C(D, m), C(O, m), and C(E, m), which deno e, espec i ely, he numbe
o n-colo pa i ions in o dis inc pa s, in o odd pa s, and in o e en pa s o a
posi i e in ege m. Also, we ake C(D, 0) = C(O, 0) = C(E, 0) = 1 as a con en ion.
The gene a ing unc ions o C(D, m), C(O, m), and C(E, m) a e gi en by
∞
X
m=0
C(D, m)qm=
∞
Y
m=1
(1 + qm)m,(2)
∞
X
m=0
C(O, m)qm=
∞
Y
m=1
1
(1 −q2m−1)2m−1,
∞
X
m=0
C(E, m)qm=
∞
Y
m=1
1
(1 −q2m)2m.
Le Ce−o(m) deno e he numbe o n-colo pa i ions o a posi i e in ege min o an
e en numbe o pa s minus he numbe o n-colo pa i ions o a posi i e in ege m
in o an odd numbe o pa s. We can easily ob ain he ollowing gene a ing unc ion
o Ce−o(m), whe e we ake Ce−o(0) = 1:
∞
X
m=0
Ce−o(m)qm=
∞
Y
m=1
1
(1 + qm)m.(3)
Now, we le Cod(m) deno e he numbe o n-colo pa i ions o mwhe ein odd
pa s a e dis inc and e en pa s a e un es ic ed and Ced(m) deno e he num-
be o n-colo pa i ions o mwhe ein e en pa s a e dis inc and odd pa s a e
un es ic ed. Again, we ake Cod(0) = 1 and Ced(0) = 1 o ob ain he ollowing
gene a ing unc ions o Cod(m) and Ced(m):
∞
X
m=0
Cod(m)qm=
∞
Y
m=1
(1 + q2m−1)2m−1
(1 −q2m)2m,
∞
X
m=0
Ced(m)qm=
∞
Y
m=1
(1 + q2m)2m
(1 −q2m−1)2m−1.
In Sec ion 2, we enume a e he pa s o n-colo pa i ions wi hou es ic ions
and in es iga e hei ela ionship wi h a ious di iso unc ions. In Sec ion 3, we
in es iga e simila ela ions while coun ing he numbe o pa s o n-colo pa i ions
wi h es ic ions.
INTEGERS: 25 (2025) 4
2. Numbe o Pa s in n-Colo Pa i ions and Di iso Func ions
We s a his sec ion by summing he numbe o pa s o all he n-colo pa i ions
in o an odd (e en) numbe o pa s which p o ides us ou i s esul .
Theo em 3. Le To(m) ( espec i ely Te(m)) deno e he sum o he numbe o pa s,
whe e he sum is aken o e all he n-colo pa i ions o a posi i e in ege min o
an odd ( espec i ely e en)numbe o pa s. Then o |q|<1,
∞
X
m=1
mqm
1∓qm=
∞
Y
m=1
(1∓qm)m
∞
X
m=1
(To(m)±Te(m))qm.
P oo . Conside ing he gene a ing unc ion o n-colo pa i ions and in oducing a
new a iable xwhich keeps ack o he numbe o pa s, we ha e
∞
X
m=1
(To(m)±Te(m))qm=d
dxx=±1(1 −xq)−1(1 −xq2)−2· · · 
=1
∞
Q
m=1
(1∓qm)m
∞
X
m=1
mqm
1∓qm.
This concludes ou p oo .
We also ha e he ollowing esul s om sequences A000593, A146076, and A000203
in he OEIS [9]:
∞
X
m=1
mqm
1 + qm=
∞
X
m=1
(2m−1)q2m−1
1−q2m−1=
∞
X
m=1
σodd(m)qm,(4)
∞
X
m=1
2mq2m
1−q2m=
∞
X
m=1
σe en(m)qm,(5)
whe e σe en(m) ( espec i ely, σodd(m)) is he sum o e en di iso s ( espec i ely,
odd di iso s) o m.
Also, ∞
X
m=1
(2m−1)q2m−1
1 + q2m−1=
∞
X
m=1
(−1)m+1σodd(m)qm
and ∞
X
m=1
mqm
1−qm=
∞
X
m=1
σ(m)qm,|q|<1,(6)
whe e σ(m) deno es he sum o di iso s o m. Now, using Equa ions (1), (2),
(4), and (6), we ge Co olla y 1and Co olla y 2 o Theo em 3wi h he help o
con olu ion p ope ies.
INTEGERS: 25 (2025) 5
Co olla y 1. Fo any posi i e in ege m,
To(m) + Te(m) =
m
X
k=1
σ(k)C(m−k).(7)
We ake an example o e i y Co olla y 1 o m= 4.
Example 1. n-colo pa i ions o 4 in o an odd numbe o pa s a e
41,42,43,44,22+ 11+ 11,21+ 11+ 11,
and in o an e en numbe o pa s a e
31+ 11,32+ 11,33+ 11,22+ 22,22+ 21,2+1+11,11+ 11+ 11+ 11.
Thus, To(4) = 10 and Te(4) = 16. We know σ(1) = 1, σ(2) = 3, σ(3) = 4, and
σ(4) = 7. Also, C(3) = 6, C(2) = 3, C(1) = 1, and C(0) = 1. On subs i u ing all
hese alues and m= 4 in (7), we see ha
To(4) + Te(4) =
4
X
k=1
σ(k)C(4 −k).
Co olla y 2. Fo any posi i e in ege m,
σodd(m) =
m
X
k=1
(To(k)−Te(k)) C(D, m −k).
In Theo em 4, we coun he numbe o e en pa s in all he n-colo pa i ions.
Theo em 4. Le Ae(m)deno e he sum o he numbe o e en pa s, whe e he sum
is aken o e all he n-colo pa i ions o a posi i e in ege m. Then o |q|<1,
∞
X
m=1
2mq2m
1−q2m=
∞
Y
m=1
(1 −qm)m
∞
X
m=1
Ae(m)qm.
P oo . P oceeding in a simila manne as in he p e ious heo em, we see ha
∞
X
m=1
Ae(m)qm=
∞
Y
m=1
1
(1 −q2m−1)2m−1
d
dxx=1
∞
Y
m=1
1
(1 −xq2m)2m
=
∞
Y
m=1
1
(1 −qm)m
∞
X
m=1
2mq2m
1−q2m.
This comple es he p oo .
As a di ec consequence o Equa ions (1) and (5), we ob ain he ollowing co ol-
la y o Theo em 4.

INTEGERS: 25 (2025) 6
Co olla y 3. Fo any posi i e in ege m,
Ae(m) =
m
X
k=1
σe en(k)C(m−k).
Theo em 5is simila o Theo em 4wi h he only di e ence being ha we now
coun he numbe o odd pa s.
Theo em 5. Le Ao(m) ep esen he sum o he numbe o odd pa s, whe e he
sum is aken o e all he n-colo pa i ions o a posi i e in ege m. Then o |q|<1,
∞
X
m=1
(2m−1)q2m−1
1−q2m−1=
∞
Y
m=1
(1 −qm)m
∞
X
m=1
Ao(m)qm.
Co olla y 4. Fo any posi i e in ege m,
Ao(m) =
m
X
k=1
σodd(k)C(m−k).
Ou nex esul is abou coun ing he numbe o pa s wi h mul iple occu ences.
Theo em 6. Fo posi i e in ege s mand , le N (m)deno e he sum o he numbe
o dis inc pa s wi h mul iplici y a leas , whe e he sum is aken o e all he n-
colo pa i ions o m. Then,
∞
X
m=1
mqm =
∞
Y
m=1
(1 −qm)m
∞
X
m=1
N (m)qm, m ≥ .
P oo . We can ob ain he gene a ing unc ion o N (m) by in oducing a a iable x
o accoun o he numbe o pa s wi h mul iplici y a leas , hen aking de i a i e
wi h espec o xand inally subs i u ing x= 1. Hence,
∞
X
m=1
N (m)qm=d
dxx=1
∞
Y
m=1 1 + qm+· · · +q( −1)m+xqm
1−qmm
=
∞
Y
m=1
1
(1 −qm)m
∞
X
m=1
mqm .
Co olla y 5. Fo posi i e in ege s mand ,
N (m) =
⌊m/ ⌋
X
k=1
kC(m− k), m ≥ .
INTEGERS: 25 (2025) 7
In Theo em 7, we coun numbe o pa s cong uen o k(mod ) o some ixed
posi i e in ege s kand .
Theo em 7. Fo ≥1and 0< k < , le C(kmod , m)deno e he sum o he
numbe o pa s cong uen o k (mod ), whe e he sum is aken o e all he n-colo
pa i ions o a posi i e in ege m. Then,
∞
X
m=0
( m +k)q m+k
1−q m+k=
∞
Y
m=1
(1 −qm)m
∞
X
m=1
C(kmod , m)qm.
P oo . In he ollowing wo a iable gene a ing unc ion, xkeeps ack o pa s
cong uen o k(mod ) in all he n-colo pa i ions:
∞
Y
i=1
i≡k(mod )
1
(1 −qi)i
∞
Y
j=1
j≡k(mod )
1
(1 −xqj)j.
By applying d
dxx=1 o he abo e exp ession, we ob ain
∞
X
m=1
C(k(mod ), m)qm=
∞
Y
i=1
i≡k(mod )
1
(1 −qi)i
∞
Y
i=1
i≡k(mod )
1
(1 −qi)i
∞
X
m=0
( m +k)q m+k
1−q m+k
=
∞
Y
m=1
1
(1 −qm)m
∞
X
m=0
( m +k)q m+k
1−q m+k.
Co olla y 6. Fo in ege s m > 0, ≥1and 0< k < ,
C(kmod , m) =
m
X
j=1

X
d+k|j
( d +k)
C(m−j).
3. Numbe o Pa s in n-Colo Pa i ions wi h Res ic ions and Di iso
Func ions
In his sec ion, we ocus on coun ing he numbe o pa s in some es ic ed n-colo
pa i ions and we s a by conside ing n-colo pa i ions in o dis inc pa s.
Theo em 8. Le Co(D, m) ( espec i ely Ce(D, m)) deno e he sum o he numbe
o pa s, whe e he sum is aken o e all he n-colo pa i ions o a posi i e in ege
min o an odd ( espec i ely e en)numbe o dis inc pa s. Then o |q|<1,
∞
X
m=1
mqm
1±qm=1
∞
Q
m=1
(1±qm)m
∞
X
m=1
(Co(D, m)±Ce(D, m)) qm.
INTEGERS: 25 (2025) 8
P oo . P oceeding in a manne simila o he p oo o Theo em 3, we ha e
∞
X
m=1
(Co(D, m)±Ce(D, m)) qm=d
dxx=±1(1 + xq)(1 + xq2)2(1 + xq3)3· · · 
=
∞
Y
m=1
(1 + xqm)mq
1 + xq +2q2
1 + xq2+· · · x=±1
=
∞
Y
m=1
(1±qm)m
∞
X
m=1
mqm
1±qm,
which gi es us he desi ed esul .
To ob ain Co olla ies 7-9 o Theo em 8, we use Equa ions (1), (2), (3), (4), (6),
and he Cauchy p oduc o wo powe se ies.
Co olla y 7. Fo any posi i e in ege m,
Co(D, m) + Ce(D, m) =
m
X
k=1
σodd(k)C(D, m −k).
Co olla y 8. Fo any posi i e in ege m,
σ(m) =
m
X
k=1
(Co(D, k)−Ce(D, k)) C(m−k).
Co olla y 9. Fo any posi i e in ege m,
σodd(m) =
m
X
k=1
C(D, k)Ce−o(m−k).
P oceeding in a simila manne as in Theo em 8, we can p o e Theo em 9.
Theo em 9. Le Q(O, m) ep esen sum o he numbe o pa s, whe e he sum is
aken o e all he n-colo pa i ions o a posi i e in ege min o odd pa s. Then
o |q|<1,
∞
X
m=1
(2m−1)q2m−1
1−q2m−1=
∞
Y
m=1
(1 −q2m−1)2m−1
∞
X
m=1
Q(O, m)qm.
Co olla y 10. Fo any posi i e in ege m,
Q(O, m) =
m
X
k=1
σodd(k)C(O, m −k).
INTEGERS: 25 (2025) 9
Simila esul s can be p o ed o he numbe o pa s in all he n-colo pa i ions
o a posi i e in ege in o e en pa s.
Ou nex heo em is ela ed o he se o n-colo pa i ions wi h dis inc odd
pa s and un es ic ed e en pa s.
Theo em 10. Le Qod(m) ep esen he sum o he numbe o pa s, whe e he sum
is aken o e all he n-colo pa i ions o a posi i e in ege mwi h dis inc odd pa s
and un es ic ed e en pa s. Then o |q|<1,
∞
X
m=1
(2m−1)q2m−1
1 + q2m−1=
∞
Y
m=1
(1 −q2m)2m
(1 + q2m−1)2m−1
∞
X
m=1
Qod(m)qm.
Co olla y 11. Fo a posi i e in ege m,
Qod(m) =
m
X
k=1

X
2d−1|k
(−1)1+k/2d−1(2d−1)
Cod(m−k).
Again, simila esul s can be p o ed o Qed(m), whe e Qed(m) ep esen s he
sum o he numbe o pa s wi h he sum aken o e all he n-colo pa i ions o a
non-nega i e in ege in o dis inc e en pa s and un es ic ed odd pa s.
Theo em 11 is also abou coun ing he numbe o pa s in n-colo pa i ions in o
dis inc pa s bu he e we coun some ixed pa .
Theo em 11. Fo j≥1, le Q(D, j, m)deno e he sum o he numbe o j’s, whe e
he sum is aken o e all he n-colo pa i ions o a posi i e in ege min o dis inc
pa s. Then o |q|<1,
∞
X
m=1
Q(D, j, m)qm=jqj
1 + qj
∞
X
m=0
C(D, m)qm.
P oo . To p o e his esul , we in oduce a a iable x o coun he o al numbe o
j’s in all he n-colo pa i ions o min o dis inc pa s. Then, we ake he de i a i e
wi h espec o xand pu x= 1. Hence,
∞
X
m=1
Q(D, j, m)qm=d
dxx=1 1 + xqjj∞
Y
i=1
i=j
(1 + qi)i
=jqj
1 + qjY
i≥1
(1 + qi)i
=jqj
1 + qj
∞
X
m=0
C(D, m)qm.