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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.
2
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
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
KEYWORD
nonn,tabf,base
AUTHOR
N. J. A. Sloane, Dec 30 2018
EXTENSIONS
More terms from Alois P. Heinz, Dec 30 2018
STATUS
approved