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# Partitions with restricted parts modulo 5

(Redirected from Partitions of 5n)

Among partitions of n we can distinguish partitions with different relationships between various c(k,5), k = 0, 1, 2, 3, 4, where c(k,m) is the number of parts in a partition of size congruent to k modulo m.

## Sequences with c(1,5) = c(4,5) and c(2,5) = c(3,5)

The number of partitions of n with c(1,5) = c(4,5) and c(2,5) = c(3,5) is zero unless n is a multiple of 5. The total number of partitions of 5n with c(1,5) = c(4,5) and c(2,5) = c(3,5) is given by A202091.

Sequences counting such restricted partitions, subject to ordering of the quantities c(1,5) = c(4,5), c(2,5) = c(3,5) (denoted by a's in the patterns), and c(0,5) (denoted by b in the patterns), are summarized in the following table:

 pattern a ? a ? b a ? b ? a b ? a ? a = = A046776 A046776 A046776 = ≤ A202085 A036884 A036884 = < A202086 A036886 A036886 ≤ = A036889 A036889 A202087 ≤ ≤ A036882 A036881 A036880 ≤ < A036887 A036885 A036883 < = A036892 A036892 A202088 < ≤ A036891 A036890 A036888 < < A036895 A036894 A036893

### Formulae

Each of the sequences can be computed as follows:

${\displaystyle a(n)=\sum _{u+v+w\leq n}\quad \sum _{m_{0}+m_{1}+m_{2}+m_{3}+m_{4}=n-u-v-w}q(m_{0},w)\cdot q(m_{1},u)\cdot q(m_{2},v)\cdot q(m_{3},v)\cdot q(m_{4},u),}$

where u stands for c(1,5) = c(4,5), v stands for c(2,5) = c(3,5), and w stands for c(0,5), and the first sum spans over u, v, w satisfying the (in)equalities imposed by a particular sequence; q(m,k) = A026820(m,k) is the number of partitions of m with at most k parts.

Correspondingly, the generating function equals

${\displaystyle \sum _{n\geq 0}a(n)\cdot x^{n}=\sum _{u,v,w}\quad x^{u+v+w}\cdot P_{u}(x)^{2}\cdot P_{v}(x)^{2}\cdot P_{w}(x),}$

where the sum again spans over u, v, w satisfying the (in)equalities imposed by a particular sequence, and ${\displaystyle P_{k}(x)=\prod _{i=1}^{k}{\tfrac {1}{1-x^{i}}}}$ is the generating function for ${\displaystyle q(\cdot ,k)}$.

### Relationship between sequences

Trivially for all n we have:

A202091(n) = A046776(n) + A202086(n) + A202088(n) + 2⋅( A036886(n) + A036892(n) + A036893(n) + A036894(n) + A036895(n) ),
A046787(n) = A046776(n) + A202086(n) + A202088(n) - A000041(n),
A202192(n) = A046776(n) + A202086(n) + A202088(n).

Furthermore, in each column of the table, there are three identities of the form:

[x =] + [x <] = [x ≤]

and three identities of the form:

[= x] + [< x] = [≤ x]

where x ∈ { "=", "<", "≤" }.

In particular, for x = "=", the first form for the first column gives the identity:

A046776(n) + A202086(n) = A202085(n) for all n.

### Sample PARI/GP code

{ q(n,k) = if(k==0,return(n==0)); if(n==1,return(k>=1)); if(k>=n,return(numbpart(n))); q(n,k-1)+q(n-k,k); }

{ A036890(n) = my(r,t);
r=0;
for(u=0,n\3, for(v=u+1,(n-u)\2, for(w=v,n-u-v,
t=n-u-v-w;
forvec(z=vector(4,i,[0,t]),
r += q(z[1],u)*q(z[2]-z[1],u)*q(z[3]-z[2],v)*q(z[4]-z[3],w)*q(t-z[4],w);
,1);
)));
r;
}


## Sequences with c(0,5) = c(2,5) and c(3,5) = c(4,5)

The total number of partitions of 5n with c(0,5) = c(2,5) and c(3,5) = c(4,5) is given by A??????.

Sequences counting such restricted partitions, subject to ordering of the quantities c(1,5) = c(4,5), c(2,5) = c(3,5) (denoted by a's in the patterns), and c(1,5) (denoted by b in the patterns), are summarized in the following table:

 pattern a ? a ? b a ? b ? a b ? a ? a = = A046776[1] A046776[1] A046776[1] = ≤ A036850 A036850 = < A036853 A036853 ≤ = A036855 A036855 ≤ ≤ A036846 A036848 A036849 ≤ < A036847 A036851 A036852 < = A036857 A036857 < ≤ A036854 A036856 A036859 < < A036858 A036860 A036861

### Formulae

Each of the sequences can be computed as follows:

${\displaystyle a(n)=\sum _{7u+7v+w\leq 5n \atop 7u+7v+w\equiv 0{\pmod {5}}}\quad \sum _{m_{0}+m_{1}+m_{2}+m_{3}+m_{4}=n-(7u+7v+w)/5}q(m_{0},u)\cdot q(m_{1},w)\cdot q(m_{2},u)\cdot q(m_{3},v)\cdot q(m_{4},v),}$

where u stands for c(0,5) = c(2,5), v stands for c(3,5) = c(4,5), and w stands for c(1,5), and the first sum spans over u, v, w additionally satisfying the (in)equalities imposed by a particular sequence.

Correspondingly, the generating function equals

${\displaystyle \sum _{n\geq 0}a(n)\cdot x^{n}=\sum _{7u+7v+w\equiv 0{\pmod {5}}}\quad x^{(7u+7v+w)/5}\cdot P_{u}(x)^{2}\cdot P_{v}(x)^{2}\cdot P_{w}(x).}$

## Sequences with c(0,5) = c(1,5) and c(2,5) = c(4,5)

The total number of partitions of 5n with c(0,5) = c(1,5) and c(2,5) = c(4,5) is given by A??????.

Sequences counting such restricted partitions, subject to ordering of the quantities c(0,5) = c(1,5), c(2,5) = c(4,5) (denoted by a's in the patterns), and c(3,5) (denoted by b in the patterns), are summarized in the following table:

 pattern a ? a ? b a ? b ? a b ? a ? a = = A046776[1] A046776[1] A046776[1] = ≤ A036866 A036866 = < A036869 A036869 ≤ = A036871 A036871 ≤ ≤ A036862 A036864 A036865 ≤ < A036863 A036867 A036868 < = A036873 A036873 < ≤ A036870 A036872 A036875 < < A036874 A036876 A036877

### Formulae

Each of the sequences can be computed as follows:

${\displaystyle a(n)=\sum _{6u+6v+3w\leq 5n \atop 6u+6v+3w\equiv 0{\pmod {5}}}\quad \sum _{m_{0}+m_{1}+m_{2}+m_{3}+m_{4}=n-(6u+6v+3w)/5}q(m_{0},u)\cdot q(m_{1},u)\cdot q(m_{2},v)\cdot q(m_{3},w)\cdot q(m_{4},v),}$

where u stands for c(0,5) = c(1,5), v stands for c(2,5) = c(4,5), and w stands for c(3,5), and the first sum spans over u, v, w additionally satisfying the (in)equalities imposed by a particular sequence.

Correspondingly, the generating function equals

${\displaystyle \sum _{n\geq 0}a(n)\cdot x^{n}=\sum _{6u+6v+3w\equiv 0{\pmod {5}}}\quad x^{(6u+6v+3w)/5}\cdot P_{u}(x)^{2}\cdot P_{v}(x)^{2}\cdot P_{w}(x).}$

## Notes

1. In fact, here a(5n) = A046776(n), while a(5n+k) = 0 for k=1,2,3,4.