A240837 Partitions as specified by composition into an even number of parts.

1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 2, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 1, 4, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 1, 4, 4, 3, 3, 1, 2, 2, 1, 1, 3, 3, 2, 5, 4, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 3, 2, 2, 4, 3, 3, 2, 1

Offset

2,4

Comments

The composition specifies the run lengths of the boundary of the Ferrers diagram of the partition.

Taking the n-th row as multiple partitions, it consists of those partitions with the first hook size (largest part plus number of parts minus 1) equal to n-1. The number of integers in this n-th row is A001792(n-2), and the row sum is A049611(n-1).

Links

Table of n, a(n) for n=2..81.

Example

For row 11, the 11th row in A240750 is 2,1,1,1. This gives us the Ferrers diagram:
* * *
* *
with boundary 2 horizontal, 1 vertical, 1 horizontal, 1 vertical. This is the diagram for partition [2,2,1].
The table starts:
[]
(none)
1
1,1; 2
1,1,1; 2,2; 3; 2,1
1,1,1,1; 2,2,2; 3,3; 2,2,1; 4; 3,1; 2,1,1; 3,2

Prog

(PARI) evil(n) = local(r=0, m=n); while(m>0, if(m%2==1, r=1-r); m\=2); n*2+r
A066099row(n) = {local(v=vector(n), j=0, k=0);
   while(n>0, k++; if(n%2==1, v[j++]=k; k=0); n\=2);
   vector(j, i, v[j-i+1])}
A240750row(n) = A066099row(evil(n))
partpath(v) = {local(j=0, n=0, m=0, r);
   forstep(k=1, #v, 2, n+=v[k]; m+=v[k+1]);
   r=vector(n);
   forstep(k=1, #v, 2, for(i=1, v[k], r[j++]=m); m-=v[k+1]);
   r}
arow(n) = partpath(A240750row(n))

Crossrefs

A240750, A066099, A125106.

Sequence in context: A051574 A029386 A277325 * A369376 A060502 A035439

Adjacent sequences: A240834 A240835 A240836 * A240838 A240839 A240840

Keyword

nonn,tabf

Author

Franklin T. Adams-Watters, Apr 13 2014

Status

approved