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A229946 Height of the peaks and the valleys in the Dyck path whose j-th ascending line segment has A141285(j) steps and whose j-th descending line segment has A194446(j) steps. 3
0, 1, 0, 2, 0, 3, 0, 2, 1, 5, 0, 3, 2, 7, 0, 2, 1, 5, 3, 6, 5, 11, 0, 3, 2, 7, 5, 9, 8, 15, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 0, 3, 2, 7, 5, 9, 8, 15, 11, 14, 13, 19, 17, 22, 21, 30, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 15, 19, 18, 25, 23, 29, 28, 33, 32, 42, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also 0 together the alternating sums of A220517.

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.

For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y].  See below:

.

.  j     Diagram 1         Partitions           Diagram 2

.      _ _ _ _ _ _ _                          _ _ _ _ _ _ _

. 15  |_ _ _ _      |      7                  _ _ _ _      |

. 14  |_ _ _ _|_    |      4+3                _ _ _ _|_    |

. 13  |_ _ _    |   |      5+2                _ _ _    |   |

. 12  |_ _ _|_ _|_  |      3+2+2              _ _ _|_ _|_  |

. 11  |_ _ _      | |      6+1                _ _ _      | |

. 10  |_ _ _|_    | |      3+3+1              _ _ _|_    | |

.  9  |_ _    |   | |      4+2+1              _ _    |   | |

.  8  |_ _|_ _|_  | |      2+2+2+1            _ _|_ _|_  | |

.  7  |_ _ _    | | |      5+1+1              _ _ _    | | |

.  6  |_ _ _|_  | | |      3+2+1+1            _ _ _|_  | | |

.  5  |_ _    | | | |      4+1+1+1            _ _    | | | |

.  4  |_ _|_  | | | |      2+2+1+1+1          _ _|_  | | | |

.  3  |_ _  | | | | |      3+1+1+1+1          _ _  | | | | |

.  2  |_  | | | | | |      2+1+1+1+1+1        _  | | | | | |

.  1  |_|_|_|_|_|_|_|      1+1+1+1+1+1+1       | | | | | | |

.

.      1 2 3 4 5 6 7

.

The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.

Also the diagram has the property that it can be transformed in a Dyck path (see example).

The height of the peaks and the valleys of the infinite Dyck path give this sequence.

Q: Is this Dyck path a fractal?

LINKS

Table of n, a(n) for n=0..84.

Omar E. Pol, Visualization of regions in a diagram for A006128

FORMULA

a(0) = 0; a(n) = a(n-1) + (-1)^(n-1)*A220517(n), n >= 1.

EXAMPLE

Illustration of initial terms (n = 0..21):

.                                                             11

.                                                             /

.                                                            /

.                                                           /

.                                   7                      /

.                                   /\                 6  /

.                     5            /  \           5    /\/

.                     /\          /    \          /\  / 5

.           3        /  \     3  /      \        /  \/

.      2    /\   2  /    \    /\/        \   2  /   3

.   1  /\  /  \  /\/      \  / 2          \  /\/

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

.  0 0   0     0           0               0

.

Note that the k-th largest peak between two valleys at height 0 is also A000041(k) and the next term is always 0.

.

Written as an irregular triangle in which row k has length 2*A187219(k), k >= 1, the sequence begins:

0,1;

0,2;

0,3;

0,2,1,5;

0,3,2,7;

0,2,1,5,3,6,5,11;

0,3,2,7,5,9,8,15;

0,2,1,5,3,6,5,11,7,12,11,15,14,22;

0,3,2,7,5,9,8,15,11,14,13,19,17,22,21,30;

0,2,1,5,3,6,5,11,7,12,11,15,14,22,15,19,18,25,23,29,28,33,32,42;

...

CROSSREFS

Column 1 is A000004. Right border gives A000041 for the positive integers.

Cf. A006128, A135010, A138137, A139582, A141285, A186412, A187219, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610.

Sequence in context: A027640 A194666 A325799 * A127460 A274021 A303711

Adjacent sequences:  A229943 A229944 A229945 * A229947 A229948 A229949

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Nov 03 2013

STATUS

approved

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Last modified February 24 02:48 EST 2020. Contains 332195 sequences. (Running on oeis4.)