OFFSET
1,2
COMMENTS
For the definition of "region" of the set of partitions of j, see A206437.
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.
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.
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
From Omar E. Pol, Aug 19 2013: (Start)
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).
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.
(End)
The equivalent sequence for compositions is A006519. - Omar E. Pol, Aug 22 2013
LINKS
Robert Price, Table of n, a(n) for n = 1..5603
Omar E. Pol, Illustration of the seven regions of 5
FORMULA
EXAMPLE
Written as an irregular triangle the sequence begins:
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, 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, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 77;
...
From Omar E. Pol, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
. _ _ _ _ _
. _ _ _ |_ _ _ _ _|
. _ _ _ _ |_ _ _| |_ _|
. _ _ |_ _ _ _| |_|
. _ _ _ |_ _| |_ _| |_|
. _ _ |_ _ _| |_| |_|
. _ |_ _| |_| |_| |_|
. |_| |_| |_| |_| |_|
.
. 1 2 3 1 5 1 7
.
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).
. _ _ _ _ _
. 7 _ _ _ |
. 6 _ _ _|_ |
. 5 _ _ | |
. 4 _ _|_ | |
. 3 _ _ | | |
. 2 _ | | | |
. 1 | | | | |
.
. 1 2 3 4 5
.
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.
. /\
. / \
. /\ / \
. / \ / \
. /\ / \ /\/ \
. /\ / \ /\/ \ / 1 \
. /\/ \/ \/ 1 \/ \
. 1 2 3 5 7
.
(End)
MATHEMATICA
lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];
A194446 = {}; l = {};
For[j = 1, j <= 30, j++,
mx = Max@lex[j][[j]]; AppendTo[l, mx];
For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
AppendTo[A194446, j - i];
];
A194446 (* Robert Price, Jul 25 2020 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Nov 26 2011
STATUS
approved