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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.

%I A322763 #15 Feb 14 2019 19:19:41
%S A322763 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,
%T A322763 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,
%U A322763 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
%N 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.
%C A322763 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.
%D A322763 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.5, Problem 73, pp. 415, 761.
%H A322763 Alois P. Heinz, <a href="/A322763/b322763.txt">Rows n = 1..24, flattened</a>
%e A322763 In compressed form (see A322762) triangle begins:
%e A322763   1,
%e A322763   1, 12,
%e A322763   1, 11, 123,
%e A322763   1, 11, 12, 112, 1234,
%e A322763   1, 11, 11, 112, 121, 1123, 12345,
%e A322763   1, 11, 11, 112, 12, 111, 1123, 123, 1212, 11234, 123456,
%e A322763   ...
%e A322763 For example, the 11 partitions of 6 are:
%e A322763 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111,
%e A322763 and applying the transformation we get:
%e A322763 1, 11, 11, 112, 12, 111, 1123, 123, 1212, 11234, 123456.
%e A322763 In the uncompressed notation the triangle begins:
%e A322763   {1},
%e A322763   {1}, {1,2},
%e A322763   {1}, {1,1}, {1,2,3},
%e A322763   {1}, {1,1}, {1,2}, {1,1,2}, {1,2,3,4},
%e A322763   {1}, {1,1}, {1,1}, {1,1,2}, {1,2,1}, {1,1,2,3}, {1,2,3,4,5},
%e A322763   ...
%p A322763 b:= (n, i)-> `if`(n=0 or i=1, [[$1..n]], [(t->
%p A322763     seq(map(x-> [$1..(t+1-j), x[]], b(n-i*(t+1-j)
%p A322763     , i-1))[], j=1..t))(iquo(n, i)), b(n, i-1)[]]):
%p A322763 T:= n-> map(x-> x[], b(n$2))[]:
%p A322763 seq(T(n), n=1..10);  # _Alois P. Heinz_, Dec 30 2018
%Y A322763 Cf. A080577, A066633, A322761, A322763.
%K A322763 nonn,tabf,base
%O A322763 1,4
%A A322763 _N. J. A. Sloane_, Dec 30 2018
%E A322763 More terms from _Alois P. Heinz_, Dec 30 2018