Partitions with restricted parts modulo 5

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:

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

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

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

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

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	A046776	A046776
= ≤		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:

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

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

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

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	A046776	A046776
= ≤		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:

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

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

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