OFFSET
1,2
COMMENTS
This can be calculated using the binary expansion of n; see the PARI program.
The n-th row consists of all partitions with hook size (maximum + number of parts - 1) equal to n.
The partitions in row n of this sequence are the conjugates of the partitions in row n of A125106 taken in reverse order.
Row n is also the reversed partial sums plus one of the n-th composition in standard order (A066099) minus one. - Gus Wiseman, Nov 07 2022
LINKS
Alois P. Heinz, Rows n = 1..12, flattened
EXAMPLE
The table starts:
1;
2; 1,1;
3; 2,2; 2,1; 1,1,1;
4; 3,3; 3,2; 2,2,2; 3,1 2,2,1 2,1,1 1,1,1,1;
...
MAPLE
b:= proc(n) option remember; `if`(n=1, [[1]],
[map(x-> map(y-> y+1, x), b(n-1))[],
map(x-> [x[], 1], b(n-1))[]])
end:
T:= n-> map(x-> x[], b(n))[]:
seq(T(n), n=1..7); # Alois P. Heinz, Sep 25 2015
MATHEMATICA
T[1] = {{1}};
T[n_] := T[n] = Join[T[n-1]+1, Append[#, 1]& /@ T[n-1]];
Array[T, 7] // Flatten (* Jean-François Alcover, Jan 25 2021 *)
PROG
(PARI) apart(n) = local(r=[1]); while(n>1, if(n%2==0, for(k=1, #r, r[k]++), r=concat(r, [1])); n\=2); r \\ Generates n-th partition.
CROSSREFS
Cf. A241596 (another version of this list of partitions), A125106, A240837, A112531, A241597 (compositions).
For other schemes to list integer partitions, please see for example A227739, A112798, A241918, A114994.
First element in each row is A008687.
Last element in each row is A065120.
Heinz numbers of rows are A253565.
Another version is A358134.
KEYWORD
nonn,tabf
AUTHOR
Franklin T. Adams-Watters, May 19 2014
STATUS
approved