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Partitions with restricted parts modulo 5

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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

Explicit formulae

Each of the sequences can be computed as follows:

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. (Max Alekseyev 22:22, 11 December 2011 (UTC))

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

Explicit formulae

Each of the sequences can be computed as follows:

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.

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

Explicit formulae

Each of the sequences can be computed as follows:

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.

Notes

  1. 1.0 1.1 1.2 1.3 1.4 1.5 In fact, here a(5n) = A046776(n), while a(5n+k) = 0 for k=1,2,3,4.