cp's OEIS Frontend

A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

%I A228350 #35 Oct 22 2013 12:38:18
%S A228350 1,2,1,1,3,2,1,1,1,2,1,1,4,3,2,2,1,1,1,1,1,2,1,1,3,2,1,1,1,2,1,1,5,4,
%T A228350 3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,2,1,1,3,2,1,1,1,2,1,1,4,3,2,2,1,1,1,1,
%U A228350 1,2,1,1,3,2,1,1,1,2,1,1,6,5,4,4,3,3
%N A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).
%C A228350 Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.
%C A228350 The equivalent sequence for integer partitions is A206437.
%F A228350 T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.
%e A228350 ---------------------------------------------------------
%e A228350 .              Diagram                Triangle
%e A228350 Compositions     of            of compositions (rows)
%e A228350 .   of 5       regions          and regions (columns)
%e A228350 ----------------------------------------------------------
%e A228350 .             _ _ _ _ _
%e A228350 .         5  |_        |                                5
%e A228350 .       1+4  |_|_      |                              1 4
%e A228350 .       2+3  |_  |     |                            2   3
%e A228350 .     1+1+3  |_|_|_    |                          1 1   3
%e A228350 .       3+2  |_    |   |                        3       2
%e A228350 .     1+2+2  |_|_  |   |                      1 2       2
%e A228350 .     2+1+2  |_  | |   |                    2   1       2
%e A228350 .   1+1+1+2  |_|_|_|_  |                  1 1   1       2
%e A228350 .       4+1  |_      | |                4               1
%e A228350 .     1+3+1  |_|_    | |              1 3               1
%e A228350 .     2+2+1  |_  |   | |            2   2               1
%e A228350 .   1+1+2+1  |_|_|_  | |          1 1   2               1
%e A228350 .     3+1+1  |_    | | |        3       1               1
%e A228350 .   1+2+1+1  |_|_  | | |      1 2       1               1
%e A228350 .   2+1+1+1  |_  | | | |    2   1       1               1
%e A228350 . 1+1+1+1+1  |_|_|_|_|_|  1 1   1       1               1
%e A228350 .
%e A228350 Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
%e A228350 .             _ _ _ _
%e A228350 .         4  |_      |                  4
%e A228350 .       1+3  |_|_    |                1   3
%e A228350 .       2+2  |_  |   |              2       2
%e A228350 .     1+1+2  |_|_|_  |            1   1       2
%e A228350 .       3+1  |_    | |          3               1
%e A228350 .     1+2+1  |_|_  | |        1   2               1
%e A228350 .     2+1+1  |_  | | |      2       1               1
%e A228350 .   1+1+1+1  |_|_|_|_|    1   1       1               1
%e A228350 .
%e A228350 Illustration of the four sections of the set of compositions of 4:
%e A228350 .                                      _ _ _ _
%e A228350 .                                     |_      |     4
%e A228350 .                                     |_|_    |   1+3
%e A228350 .                                     |_  |   |   2+2
%e A228350 .                       _ _ _         |_|_|_  | 1+1+2
%e A228350 .                      |_    |   3          | |     1
%e A228350 .             _ _      |_|_  | 1+2          | |     1
%e A228350 .     _      |_  | 2       | |   1          | |     1
%e A228350 .    |_| 1     |_| 1       |_|   1          |_|     1
%e A228350 .
%e A228350 .
%e A228350 Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
%e A228350 --------------------------------------------------------
%e A228350 \j:  1      2    3        4     5      6    7          8
%e A228350 k
%e A228350 --------------------------------------------------------
%e A228350 .  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _
%e A228350 1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4
%e A228350 2        |_|1        |_  |2         |_|1        |_    |3
%e A228350 3                      | |1                       |   |2
%e A228350 4                      |_|1                       |_  |2
%e A228350 5                                                   | |1
%e A228350 6                                                   | |1
%e A228350 7                                                   | |1
%e A228350 8                                                   |_|1
%e A228350 .
%e A228350 Triangle begins:
%e A228350 1;
%e A228350 2,1;
%e A228350 1;
%e A228350 3,2,1,1;
%e A228350 1;
%e A228350 2,1;
%e A228350 1;
%e A228350 4,3,2,2,1,1,1,1;
%e A228350 1;
%e A228350 2,1;
%e A228350 1;
%e A228350 3,2,1,1;
%e A228350 1;
%e A228350 2,1;
%e A228350 1;
%e A228350 5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
%e A228350 ...
%e A228350 .
%e A228350 Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
%e A228350 [1];
%e A228350 [2,1];
%e A228350 [1],[3,2,1,1];
%e A228350 [1],[2,1],[1],[4,3,2,2,1,1,1,1];
%e A228350 [1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];
%Y A228350 Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.
%Y A228350 Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.
%K A228350 nonn,tabf
%O A228350 1,2
%A A228350 _Omar E. Pol_, Aug 20 2013