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 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). 7
 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, 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, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 6, 5, 4, 4, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1. The equivalent sequence for integer partitions is A206437. LINKS FORMULA T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1. EXAMPLE --------------------------------------------------------- .              Diagram                Triangle Compositions     of            of compositions (rows) .   of 5       regions          and regions (columns) ---------------------------------------------------------- .             _ _ _ _ _ .         5  |_        |                                5 .       1+4  |_|_      |                              1 4 .       2+3  |_  |     |                            2   3 .     1+1+3  |_|_|_    |                          1 1   3 .       3+2  |_    |   |                        3       2 .     1+2+2  |_|_  |   |                      1 2       2 .     2+1+2  |_  | |   |                    2   1       2 .   1+1+1+2  |_|_|_|_  |                  1 1   1       2 .       4+1  |_      | |                4               1 .     1+3+1  |_|_    | |              1 3               1 .     2+2+1  |_  |   | |            2   2               1 .   1+1+2+1  |_|_|_  | |          1 1   2               1 .     3+1+1  |_    | | |        3       1               1 .   1+2+1+1  |_|_  | | |      1 2       1               1 .   2+1+1+1  |_  | | | |    2   1       1               1 . 1+1+1+1+1  |_|_|_|_|_|  1 1   1       1               1 . 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: .             _ _ _ _ .         4  |_      |                  4 .       1+3  |_|_    |                1   3 .       2+2  |_  |   |              2       2 .     1+1+2  |_|_|_  |            1   1       2 .       3+1  |_    | |          3               1 .     1+2+1  |_|_  | |        1   2               1 .     2+1+1  |_  | | |      2       1               1 .   1+1+1+1  |_|_|_|_|    1   1       1               1 . Illustration of the four sections of the set of compositions of 4: .                                      _ _ _ _ .                                     |_      |     4 .                                     |_|_    |   1+3 .                                     |_  |   |   2+2 .                       _ _ _         |_|_|_  | 1+1+2 .                      |_    |   3          | |     1 .             _ _      |_|_  | 1+2          | |     1 .     _      |_  | 2       | |   1          | |     1 .    |_| 1     |_| 1       |_|   1          |_|     1 . . Illustration of initial terms. The parts of the eight regions of the set of compositions of 4: -------------------------------------------------------- \j:  1      2    3        4     5      6    7          8 k -------------------------------------------------------- .  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _ 1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4 2        |_|1        |_  |2         |_|1        |_    |3 3                      | |1                       |   |2 4                      |_|1                       |_  |2 5                                                   | |1 6                                                   | |1 7                                                   | |1 8                                                   |_|1 . Triangle begins: 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; 5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1; ... . 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): [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],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1]; CROSSREFS Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712. Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526. Sequence in context: A030312 A030321 A030305 * A220482 A084580 A263633 Adjacent sequences:  A228347 A228348 A228349 * A228351 A228352 A228353 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Aug 20 2013 STATUS approved

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Last modified January 19 08:18 EST 2022. Contains 350464 sequences. (Running on oeis4.)