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
Examples
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},
...
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
Programs
-
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
Extensions
More terms from Alois P. Heinz, Dec 30 2018
Comments