A228110 Height after n-th step of the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, n >= 0, k >= 1.

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Author

Omar E. Pol, Aug 10 2013

Keywords

• nonn
• tabf

Comments

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 sequence gives the height of the infinite Dyck path after n-th step.

The absolute values of the first differences give A000012.

For the height of the peaks and the valleys in the infinite Dyck path see A229946.

Q: Is this infinite Dyck path a fractal?

Examples

			Illustration of initial terms (n = 1..59):
.
11 ...........................................................
.                                                            /
.                                                           /
.                                                          /
7 ..................................                      /
.                                  /\                    /
5 ....................            /  \                /\/
.                    /\          /    \          /\  /
3 ..........        /  \        /      \        /  \/
2 .....    /\      /    \    /\/        \      /
1 ..  /\  /  \  /\/      \  /            \  /\/
.  /\/  \/    \/          \/              \/
.
Note that the j-th largest peak between two valleys at height 0 is also the partition number A000041(j).
Written as an irregular triangle in which row k has length 2*A138137(k), the sequence begins:
0,1;
0,1,2,1;
0,1,2,3,2,1;
0,1,2,1,2,3,4,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,4,3,2,1;
0,1,2,1,2,3,4,5,4,3,4,5,6,5,6,7,8,9,10,11,10,9,8,7,6,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,6,7,8,9,8,9,10,11,12,13,14,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1;
...
		

Links

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

Crossrefs

Column 1 is A000004. Both column 2 and the right border are in A000012. Both columns 3 and 5 are in A007395.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A229946.