A228527 -id:A228527 - cp's OEIS frontend

A228526 Triangle read by rows: T(n,k) = sum of all parts of size k in all compositions (ordered partitions) of n.

1, 2, 2, 5, 4, 3, 12, 10, 6, 4, 28, 24, 15, 8, 5, 64, 56, 36, 20, 10, 6, 144, 128, 84, 48, 25, 12, 7, 320, 288, 192, 112, 60, 30, 14, 8, 704, 640, 432, 256, 140, 72, 35, 16, 9, 1536, 1408, 960, 576, 320, 168, 84, 40, 18, 10, 3328, 3072, 2112, 1280, 720

Offset: 1

Omar E. Pol, Aug 28 2013

The equivalent sequence for partitions is A138785, see the first comment there.

			T(4,2) = 10 because there are 5 parts of size 2 in all compositions of 4, T(4,2) = 5*2 = 10 (see below):
---------------------------------------------------------
. Compositions                   Parts      Sum of parts
.     of 4        Diagram      of size 2     of size 2
---------------------------------------------------------
.                 _ _ _ _
.   1+1+1+1      |_| | | |         0             0
.     2+1+1      |_ _| | |         1             2
.     1+2+1      |_|   | |         1             2
.       3+1      |_ _ _| |         0             0
.     1+1+2      |_| |   |         1             2
.       2+2      |_ _|   |         2             4
.       1+3      |_|     |         0             0
.         4      |_ _ _ _|         0             0
.                                -----        ------
.                           Total  5            10
.
Triangle begins:
1;
2,       2;
5,       4,    3;
12,     10,    6,    4;
28,     24,   15,    8,   5;
64,     56,   36,   20,  10,   6;
144,   128,   84,   48,  25,  12,   7;
320,   288,  192,  112,  60,  30,  14,  8;
704,   640,  432,  256, 140,  72,  35, 16,  9;
1536, 1408,  960,  576, 320, 168,  84, 40, 18, 10;
3328, 3072, 2112, 1280, 720, 384, 196, 96, 45, 20, 11;
...

Column k is k*A045623. Row sums give A001787, n >= 1. Right border gives A000027.

Cf. A001792, A011782, A138785, A221876, A228525, A228527.

T(n,k) = k*A045623(n-k) = k*A221876(n,k), n >=1, 1<=k<=n.

A228524 Triangle read by rows: T(n,k) = total number of occurrences of parts k in the n-th section of the set of compositions (ordered partitions) of any integer >= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 7, 3, 1, 1, 16, 7, 3, 1, 1, 36, 16, 7, 3, 1, 1, 80, 36, 16, 7, 3, 1, 1, 176, 80, 36, 16, 7, 3, 1, 1, 384, 176, 80, 36, 16, 7, 3, 1, 1, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1, 1792, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1, 3840, 1792, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1

Offset: 1

Omar E. Pol, Aug 26 2013

Here, define "n-th section of the set of compositions of any integer >= n" to be the set formed by all parts that occur as a result of taking all compositions (ordered partitions) of n and then remove all parts of the compositions of n-1, if n >= 1. Hence the n-th section of the set of compositions of any integer >= n is also the last section of the set of compositions of n. Note that by definition the ordering of compositions is not relevant. For the visualization of the sections here we use a dissection of the diagram of compositions of n in colexicographic order, see example.
The equivalent sequence for partitions is A182703.
Row n lists the first n terms of A045891 in decreasing order.

			Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4, also the first four sections of the set of compositions of any integer >= 4:
.
.            1      2        3          4
.            _      _        _          _
.           |_|   _| |      | |        | |
.                |_ _|   _ _| |        | |
.                       |_|   |        | |
.                       |_ _ _|   _ _ _| |
.                                |_| |   |
.                                |_ _|   |
.                                |_|     |
.                                |_ _ _ _|
.
For n = 4 and k = 2, T(4,2) = 3 because there are 3 parts of size 2 in all compositions of 4, see below:
--------------------------------------------------------
.                         The last section    Number of
.   Composition of 4        of the set of      parts of
.                         compositions of 4     size k
. --------------------   -------------------
.            Diagram             Diagram    k = 1 2 3 4
. ------------------------------------------------------
.            _ _ _ _                    _
.  1+1+1+1  |_| | | |         1        | |      1 0 0 0
.    2+1+1  |_ _| | |         1        | |      1 0 0 0
.    1+2+1  |_|   | |         1        | |      1 0 0 0
.      3+1  |_ _ _| |         1   _ _ _| |      1 0 0 0
.    1+1+2  |_| |   |     1+1+2  |_| |   |      2 1 0 0
.      2+2  |_ _|   |       2+2  |_ _|   |      0 2 0 0
.      1+3  |_|     |       1+3  |_|     |      1 0 1 0
.        4  |_ _ _ _|         4  |_ _ _ _|      0 0 0 1
.                                              ---------
.                      Column sums give row 4:  7,3,1,1
.
Triangle begins:
1;
1,       1;
3,       1,   1;
7,       3,   1,   1;
16,      7,   3,   1,  1;
36,     16,   7,   3,  1,  1;
80,     36,  16,   7,  3,  1,   1;
176,    80,  36,  16,  7,  3,   1,  1;
384,   176,  80,  36, 16,  7,   3,  1,  1;
832,   384, 176,  80, 36, 16,   7,  3,  1,  1;
1792,  832, 384, 176, 80, 36,  16,  7,  3,  1, 1;
3840, 1792, 832, 384,176, 80,  36, 16,  7,  3, 1, 1;
8192, 3840,1792, 832,384,176,  80, 36, 16,  7, 3, 1, 1;
...

Row sums give A045623. Every column gives A045891.

Cf. A011782, A182703, A221876, A228350, A228366, A228367, A228370, A228526, A228527.

T(n,k) = A045891(n-k), n >= 1, 1<=k<=n.

cp's OEIS Frontend

A228526 Triangle read by rows: T(n,k) = sum of all parts of size k in all compositions (ordered partitions) of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula