0,5

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?

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

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;

...

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.

Sequence in context: A360099 A106509 A324692 * A329116 A255175 A196199

Adjacent sequences: A228107 A228108 A228109 * A228111 A228112 A228113

nonn,tabf

Omar E. Pol, Aug 10 2013

approved