A194446 - OEIS

%I #84 Jul 25 2020 11:58:46

%S 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,

%T 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,

%U 1,1,12,1,2,1,4,1,2,1,1,77,1,2,1

%N Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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

%C 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.

%C 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.

%C 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

%C From _Omar E. Pol_, Aug 19 2013: (Start)

%C 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).

%C 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.

%C (End)

%C The equivalent sequence for compositions is A006519. - _Omar E. Pol_, Aug 22 2013

%H Robert Price, <a href="/A194446/b194446.txt">Table of n, a(n) for n = 1..5603</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>

%F a(n) = A141285(n) - A194447(n). - _Omar E. Pol_, Mar 04 2012

%e Written as an irregular triangle the sequence begins:

%e   1;

%e   2;

%e   3;

%e   1, 5;

%e   1, 7;

%e   1, 2, 1, 11;

%e   1, 2, 1, 15;

%e   1, 2, 1,  4, 1, 1, 22;

%e   1, 2, 1,  4, 1, 2,  1, 30;

%e   1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 42;

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

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

%e   ...

%e From _Omar E. Pol_, Aug 18 2013: (Start)

%e Illustration of initial terms (first seven regions):

%e .                                             _ _ _ _ _

%e .                                     _ _ _  |_ _ _ _ _|

%e .                           _ _ _ _  |_ _ _|       |_ _|

%e .                     _ _  |_ _ _ _|                 |_|

%e .             _ _ _  |_ _|     |_ _|                 |_|

%e .       _ _  |_ _ _|             |_|                 |_|

%e .   _  |_ _|     |_|             |_|                 |_|

%e .  |_|   |_|     |_|             |_|                 |_|

%e .

%e .   1     2       3     1         5       1           7

%e .

%e 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).

%e .      _ _ _ _ _

%e .  7   _ _ _    |

%e .  6   _ _ _|_  |

%e .  5   _ _    | |

%e .  4   _ _|_  | |

%e .  3   _ _  | | |

%e .  2   _  | | | |

%e .  1    | | | | |

%e .

%e .      1 2 3 4 5

%e .

%e 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.

%e .                                    /\

%e .                                   /  \

%e .                      /\          /    \

%e .                     /  \        /      \

%e .            /\      /    \    /\/        \

%e .       /\  /  \  /\/      \  / 1          \

%e .    /\/  \/    \/ 1        \/              \

%e .     1   2     3           5               7

%e .

%e (End)

%t lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];

%t A194446 = {}; l = {};

%t For[j = 1, j <= 30, j++,

%t   mx = Max@lex[j][[j]]; AppendTo[l, mx];

%t   For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];

%t   AppendTo[A194446, j - i];

%t   ];

%t A194446   (* _Robert Price_, Jul 25 2020 *)

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

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

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Nov 26 2011