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 A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j). 62
 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, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
 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 a(n) = A141285(n) - A194447(n). - Omar E. Pol, Mar 04 2012 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 Row j has length A187219(j). Right border gives A000041, j >= 1. Records give A000041, j >= 1. Row sums give A138137. Cf. A002865, A006128, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194447. Sequence in context: A080305 A220137 A053815 * A251758 A250480 A326584 Adjacent sequences:  A194443 A194444 A194445 * A194447 A194448 A194449 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Nov 26 2011 STATUS approved

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Last modified April 20 06:59 EDT 2021. Contains 343125 sequences. (Running on oeis4.)