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A322763
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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.
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2
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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
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listen;
history;
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internal format)
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OFFSET
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1,4
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COMMENTS
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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.
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.5, Problem 73, pp. 415, 761.
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LINKS
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EXAMPLE
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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},
...
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MAPLE
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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))[]:
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CROSSREFS
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KEYWORD
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nonn,tabf,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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