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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..78.

FORMULA

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

EXAMPLE

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;

...

CROSSREFS

Row sums give A045623. Every column gives A045891.

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

Sequence in context: A332538 A071812 A248133 * A116407 A309402 A135288

Adjacent sequences:  A228521 A228522 A228523 * A228525 A228526 A228527

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Aug 26 2013

STATUS

approved

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Last modified July 28 10:29 EDT 2021. Contains 346326 sequences. (Running on oeis4.)