A242628 Irregular table enumerating partitions; n-th row has partitions in previous row with each part incremented, followed by partitions in previous row with an additional part of size 1.

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, 5, 4, 4, 4, 3, 3, 3, 3, 4, 2, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2, 4, 1, 3, 3, 1, 3, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 5, 5, 4, 4, 4, 4, 5, 3, 4, 4, 3, 4, 3, 3, 3, 3, 3, 3, 5, 2

Offset: 1

Author

Franklin T. Adams-Watters, May 19 2014

Keywords

• nonn
• tabf

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

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..12, flattened
• Gus Wiseman, Statistics, classes, and transformations of standard compositions

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.

Cf. A000120, A029837, A029931, A030303, A066099, A070939, A253566, A355536, A358135, A358136.

Programs

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 *)

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.