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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.
Contents
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:
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
where the sum again spans over u, v, w satisfying the (in)equalities imposed by a particular sequence, and is the generating function for .
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:
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:
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
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:
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
Notes
- ↑ ^{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.