|
%I #35 Oct 22 2013 12:38:18
%S 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 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 1,2,1,1,3,2,1,1,1,2,1,1,6,5,4,4,3,3
%N 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 Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.
%C The equivalent sequence for integer partitions is A206437.
%F T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.
%e ---------------------------------------------------------
%e . Diagram Triangle
%e Compositions of of compositions (rows)
%e . of 5 regions and regions (columns)
%e ----------------------------------------------------------
%e . _ _ _ _ _
%e . 5 |_ | 5
%e . 1+4 |_|_ | 1 4
%e . 2+3 |_ | | 2 3
%e . 1+1+3 |_|_|_ | 1 1 3
%e . 3+2 |_ | | 3 2
%e . 1+2+2 |_|_ | | 1 2 2
%e . 2+1+2 |_ | | | 2 1 2
%e . 1+1+1+2 |_|_|_|_ | 1 1 1 2
%e . 4+1 |_ | | 4 1
%e . 1+3+1 |_|_ | | 1 3 1
%e . 2+2+1 |_ | | | 2 2 1
%e . 1+1+2+1 |_|_|_ | | 1 1 2 1
%e . 3+1+1 |_ | | | 3 1 1
%e . 1+2+1+1 |_|_ | | | 1 2 1 1
%e . 2+1+1+1 |_ | | | | 2 1 1 1
%e . 1+1+1+1+1 |_|_|_|_|_| 1 1 1 1 1
%e .
%e 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 . _ _ _ _
%e . 4 |_ | 4
%e . 1+3 |_|_ | 1 3
%e . 2+2 |_ | | 2 2
%e . 1+1+2 |_|_|_ | 1 1 2
%e . 3+1 |_ | | 3 1
%e . 1+2+1 |_|_ | | 1 2 1
%e . 2+1+1 |_ | | | 2 1 1
%e . 1+1+1+1 |_|_|_|_| 1 1 1 1
%e .
%e Illustration of the four sections of the set of compositions of 4:
%e . _ _ _ _
%e . |_ | 4
%e . |_|_ | 1+3
%e . |_ | | 2+2
%e . _ _ _ |_|_|_ | 1+1+2
%e . |_ | 3 | | 1
%e . _ _ |_|_ | 1+2 | | 1
%e . _ |_ | 2 | | 1 | | 1
%e . |_| 1 |_| 1 |_| 1 |_| 1
%e .
%e .
%e Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
%e --------------------------------------------------------
%e \j: 1 2 3 4 5 6 7 8
%e k
%e --------------------------------------------------------
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e 1 |_|1 |_ |2 |_|1 |_ |3 |_|1 |_ |2 |_|1 |_ |4
%e 2 |_|1 |_ |2 |_|1 |_ |3
%e 3 | |1 | |2
%e 4 |_|1 |_ |2
%e 5 | |1
%e 6 | |1
%e 7 | |1
%e 8 |_|1
%e .
%e Triangle begins:
%e 1;
%e 2,1;
%e 1;
%e 3,2,1,1;
%e 1;
%e 2,1;
%e 1;
%e 4,3,2,2,1,1,1,1;
%e 1;
%e 2,1;
%e 1;
%e 3,2,1,1;
%e 1;
%e 2,1;
%e 1;
%e 5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
%e ...
%e .
%e 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 [1];
%e [2,1];
%e [1],[3,2,1,1];
%e [1],[2,1],[1],[4,3,2,2,1,1,1,1];
%e [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 Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.
%Y 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 nonn,tabf
%O 1,2
%A _Omar E. Pol_, Aug 20 2013
|
