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%I #184 Dec 30 2022 06:33:22
%S 1,2,1,3,1,1,2,4,2,1,1,1,3,5,2,1,1,1,1,1,2,4,2,3,6,3,2,2,1,1,1,1,1,1,
%T 1,3,5,2,4,7,3,2,2,1,1,1,1,1,1,1,1,1,1,1,2,4,2,3,6,3,2,2,5,4,8,4,3,2,
%U 2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Triangle read by rows: T(j,k) is the k-th part of the j-th region of the set of partitions of n, if 1 <= j <= A000041(n).
%C Here the j-th "region" of the set of partitions of n (or more simply the j-th "region" of n) is defined to be the first h elements of the sequence formed by the smallest parts in nonincreasing order of the partitions of the largest part of the j-th partition of n, with the list of partitions in lexicographic ordering, where h = j - i, and i is the index of the previous partition of n whose largest part is greater than the largest part of the j-th partition of n, or i = 0 if such previous largest part does not exist. The largest part of the j-th region of n is A141285(j) and the number of parts is h = A194446(j).
%C Some properties of the regions of n:
%C - The number of regions of n equals the number of partitions of n (see A000041).
%C - The set of regions of n contain the sets of regions of all positive integers previous to n.
%C - The first j regions of n are also first j regions of all integers greater than n.
%C - The sums of all largest parts of all regions of n equals the total number of parts of all regions of n. See A006128(n).
%C - If T(j,1) is a record in the sequence then the leading diagonals of triangle formed by the first j rows give the partitions of n (see example).
%C - The rank of a region is the largest part minus the number of parts (see A194447).
%C - The sum of all ranks of the regions of n is equal to zero.
%C How to make a diagram of the regions and partitions of n: in the first quadrant of the square grid we draw a horizontal line {[0, 0],[n, 0]} of length n. Then we draw a vertical line {[0, 0],[0, p(n)]} of length p(n) where p(n) is the number of partitions of n. Then, for j = 1..p(n), we draw a horizontal line {[0, j],[g, j] where g = A141285(j) is the largest part of the j-th partition of n, with the list of partitions in lexicographic order. Then, for n = 1 .. p(n), we draw a vertical line from the point [g,j] up to intercept the next segment in a lower row with respect to the axis "y". So we have a number of closed regions. Then we divide each region of n in horizontal rectangles with shorter sides = 1. We can see that in the original rectangle of area n*p(n) each row contains a set of rectangles whose areas are equal to the parts of one of the partitions of n. Then each region of n is labeled according to the position of its largest part on axis "y". Note that each region of n is similar to a mirror version of the Young diagram of one of the partitions of s, where s is the sum of all parts of the region. See the illustrations of the seven regions of 5 in the Links section.
%C Note that if row j of triangle contains parts of size 1 then the parts of row j are the smallest parts of all partitions of T(j,1), (see A046746), and also T(j,1) is a record in the sequence and also j is the number of partitions of T(j,1), (see A000041). Otherwise, if row j does not contain parts of size 1 then the parts of row j are the emergent parts of the next record in the sequence (see A183152). Row j is also the partition of A186412(j).
%C Also triangle read by rows in which row r lists the parts of the last section of the set of partitions of r, ordered by regions, such that the previous parts to the part of size r are the emergent parts of the partitions of r (see A138152) and the rest are the smallest parts of the partitions of r (see example). - _Omar E. Pol_, Apr 28 2012
%H Robert Price, <a href="/A206437/b206437.txt">Table of n, a(n) for n = 1..321</a>, first 75 regions.
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa308.jpg">Illustration of initial terms, regions = 1..77 (2D view)</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa312.jpg">Illustration of initial terms, regions = 1..30 (3D view)</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a diagram for A006128</a>
%H Robert Price, <a href="/A206437/a206437.txt">Mathematica program to draw diagram up to n=28</a>
%e -----------------------------------------
%e Region Triangle
%e j of parts
%e -----------------------------------------
%e 1 1;
%e 2 2,1;
%e 3 3,1,1;
%e 4 2;
%e 5 4,2,1,1,1;
%e 6 3;
%e 7 5,2,1,1,1,1,1;
%e 8 2;
%e 9 4,2;
%e 10 3;
%e 11 6,3,2,2,1,1,1,1,1,1,1;
%e 12 3;
%e 13 5,2;
%e 14 4;
%e 15 7,3,2,2,1,1,1,1,1,1,1,1,1,1,1;
%e .
%e The rotated triangle shows each row as a partition:
%e .
%e . 7
%e . 4 3
%e . 5 2
%e . 3 2 2
%e . 6 1
%e . 3 3 1
%e . 4 2 1
%e . 2 2 2 1
%e . 5 1 1
%e . 3 2 1 1
%e . 4 1 1 1
%e . 2 2 1 1 1
%e . 3 1 1 1 1
%e . 2 1 1 1 1 1
%e 1 1 1 1 1 1 1
%e .
%e Alternative interpretation of this sequence:
%e Triangle read by rows in which row r lists the parts of the last section of the set of partitions of r ordered by regions (see comments):
%e [1];
%e [2,1];
%e [3,1,1];
%e [2],[4,2,1,1,1];
%e [3],[5,2,1,1,1,1,1];
%e [2],[4,2],[3],[6,3,2,2,1,1,1,1,1,1,1];
%e [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1];
%t lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];
%t reg = {}; l = {};
%t For[j = 1, j <= 22, j++,
%t mx = Max@lex[j][[j]]; AppendTo[l, mx];
%t For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
%t AppendTo[reg, Take[Reverse[First /@ lex[mx]], j - i]];
%t ];
%t Flatten@reg (* _Robert Price_, Apr 21 2020, revised Jul 24 2020 *)
%Y Positive integers in A193870. Column 1 is A141285. Row j has length A194446(j). Row sums give A186412. Records are A000027.
%Y Cf. A000041, A046746, A135010, A138121, A182699, A182703, A182709, A183152, A186114, A187219, A194436-A194439, A194447, A194448, A196025, A198381.
%K nonn,tabf,look
%O 1,2
%A _Omar E. Pol_, Feb 14 2012
%E Further edited by _Omar E. Pol_, Mar 31 2012, Jan 27 2013
%E Minor edits by _Omar E. Pol_, Apr 23 2020
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