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%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
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