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

1,2

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

Gus Wiseman, Statistics, classes, and transformations of standard compositions

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.

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

Sequence in context: A208101 A131333 A120643 * A111867 A326036 A133776

Adjacent sequences: A242625 A242626 A242627 * A242629 A242630 A242631

Keyword

nonn,tabf

Author

Franklin T. Adams-Watters, May 19 2014

Status

approved