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A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j). 60
1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 1, 4, 1, 1, 22, 1, 2, 1, 4, 1, 2, 1, 30, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 42, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 56, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 77, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the definition of "region" of the set of partitions of j, see A206437.

a(n) is also the number of positive integers in the n-th row of triangle A186114. a(n) is also the number of positive integers in the n-th row of triangle A193870.

Also triangle read by rows: T(j,k) = number of parts in the k-th region of the last section of the set of partitions of j. See example. For more information see A135010.

a(n) is also the length of the n-th vertical line segment in the minimalist diagram of regions and partitions. The length of the n-th horizontal line segment is A141285(n). See also A194447. - Omar E. Pol, Mar 04 2012

From Omar E. Pol, Aug 19 2013: (Start)

In order to construct this sequence with a cellular automaton we use the following rules: We start in the first quadrant of the square grid with no toothpicks. At stage n we place A141285(n) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the point (0, n). Then we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. a(n) is the number of toothpicks in vertical direction added at n-th stage (see example section and A139250, A225600, A225610).

a(n) is also the length of the n-th descendent line segment in an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). See Example section. For more information see A211978, A220517, A225600.

(End)

The equivalent sequence for compositions is A006519. - Omar E. Pol, Aug 22 2013

LINKS

Table of n, a(n) for n=1..80.

Omar E. Pol, Illustration of the seven regions of 5

FORMULA

a(n) = A141285(n) - A194447(n). - Omar E. Pol, Mar 04 2012

EXAMPLE

Written as an irregular triangle the sequence begins:

  1;

  2;

  3;

  1, 5;

  1, 7;

  1, 2, 1, 11;

  1, 2, 1, 15;

  1, 2, 1,  4, 1, 1, 22;

  1, 2, 1,  4, 1, 2,  1, 30;

  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 42;

  1, 2, 1,  4, 1, 2,  1,  8, 1, 1, 3,  1, 1, 56;

  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 12, 1,  2, 1, 4, 1, 2, 1, 1, 77;

  ...

From Omar E. Pol, Aug 18 2013: (Start)

Illustration of initial terms (first seven regions):

.                                             _ _ _ _ _

.                                     _ _ _  |_ _ _ _ _|

.                           _ _ _ _  |_ _ _|       |_ _|

.                     _ _  |_ _ _ _|                 |_|

.             _ _ _  |_ _|     |_ _|                 |_|

.       _ _  |_ _ _|             |_|                 |_|

.   _  |_ _|     |_|             |_|                 |_|

.  |_|   |_|     |_|             |_|                 |_|

.

.   1     2       3     1         5       1           7

.

The next figure shows a minimalist diagram of the first seven regions. The n-th horizontal line segment has length A141285(n). a(n) is the length of the n-th vertical line segment, which is the vertical line segment ending in row n (see also A225610).

.      _ _ _ _ _

.  7   _ _ _    |

.  6   _ _ _|_  |

.  5   _ _    | |

.  4   _ _|_  | |

.  3   _ _  | | |

.  2   _  | | | |

.  1    | | | | |

.

.      1 2 3 4 5

.

Illustration of initial terms from an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). a(n) is the length of the n-th descendent line segment.

.                                    /\

.                                   /  \

.                      /\          /    \

.                     /  \        /      \

.            /\      /    \    /\/        \

.       /\  /  \  /\/      \  / 1          \

.    /\/  \/    \/ 1        \/              \

.     1   2     3           5               7

.

(End)

CROSSREFS

Row j has length A187219(j). Right border gives A000041, j >= 1. Records give A000041, j >= 1. Row sums give A138137.

Cf. A002865, A006128, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194447.

Sequence in context: A080305 A220137 A053815 * A251758 A250480 A166333

Adjacent sequences:  A194443 A194444 A194445 * A194447 A194448 A194449

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Nov 26 2011

STATUS

approved

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Last modified May 25 19:25 EDT 2019. Contains 323576 sequences. (Running on oeis4.)