A322763 Irregular triangle read by rows: to get row n, take partitions of n ordered as in A080577, and in each partition, change each j-th occurrence of k to j; use uncompressed notation as in A080577.

1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 1, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 4, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7

Offset

1,4

Comments

The compressed form seems easier to understand. This is A322762 but with each partition, after it has been transformed, written as the string of its parts.

References

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.5, Problem 73, pp. 415, 761.

Links

Alois P. Heinz, Rows n = 1..24, flattened

Example

In compressed form (see A322762) triangle begins:
  1,
  1, 12,
  1, 11, 123,
  1, 11, 12, 112, 1234,
  1, 11, 11, 112, 121, 1123, 12345,
  1, 11, 11, 112, 12, 111, 1123, 123, 1212, 11234, 123456,
  ...
For example, the 11 partitions of 6 are:
6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111,
and applying the transformation we get:
1, 11, 11, 112, 12, 111, 1123, 123, 1212, 11234, 123456.
In the uncompressed notation the triangle begins:
  {1},
  {1}, {1,2},
  {1}, {1,1}, {1,2,3},
  {1}, {1,1}, {1,2}, {1,1,2}, {1,2,3,4},
  {1}, {1,1}, {1,1}, {1,1,2}, {1,2,1}, {1,1,2,3}, {1,2,3,4,5},
  ...

Maple

b:= (n, i)-> `if`(n=0 or i=1, [[$1..n]], [(t->
    seq(map(x-> [$1..(t+1-j), x[]], b(n-i*(t+1-j)
    , i-1))[], j=1..t))(iquo(n, i)), b(n, i-1)[]]):
T:= n-> map(x-> x[], b(n$2))[]:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 30 2018

Crossrefs

Cf. A080577, A066633, A322761, A322763.

Sequence in context: A026791 A080576 A321744 * A213211 A294775 A369929

Adjacent sequences: A322760 A322761 A322762 * A322764 A322765 A322766

Keyword

nonn,tabf,base

Author

N. J. A. Sloane, Dec 30 2018

Extensions

More terms from Alois P. Heinz, Dec 30 2018

Status

approved