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%I #169 Apr 25 2024 13:57:05
%S 1,1,2,1,1,3,1,1,1,2,2,4,1,1,1,1,1,2,3,5,1,1,1,1,1,1,1,2,2,2,2,4,3,3,
%T 6,1,1,1,1,1,1,1,1,1,1,1,2,2,3,2,5,3,4,7,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,2,2,2,2,2,2,4,2,3,3,2,6,3,5,4,4,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.
%C This is the original sequence of a large number of sequences connected with the section model of partitions.
%C Here "the n-th section of the set of partitions of any integer greater than or equal to n" (hence "the last section of the set of partitions of n") is defined to be the set formed by all parts that occur as a result of taking all partitions of n and then removing all parts of the partitions of n-1. For integers greater than 1 the structure of a section has two main areas: the head and tail. The head is formed by the partitions of n that do not contain 1 as a part. The tail is formed by A000041(n-1) partitions of 1. The set of partitions of n contains the sets of partitions of the previous numbers. The section model of partitions has several versions according with the ordering of the partitions or with the representation of the sections. In this sequence we use the ordering of A026791.
%C The section model of partitions can be interpreted as a table of partitions. See also A138121. - _Omar E. Pol_, Nov 18 2009
%C It appears that the versions of the model show an overlapping of sections and subsections of the numbers congruent to k mod m into parts >= m. For example:
%C First generation (the main table):
%C Table 1.0: Partitions of integers congruent to 0 mod 1 into parts >= 1.
%C Second generation:
%C Table 2.0: Partitions of integers congruent to 0 mod 2 into parts >= 2.
%C Table 2.1: Partitions of integers congruent to 1 mod 2 into parts >= 2.
%C Third generation:
%C Table 3.0: Partitions of integers congruent to 0 mod 3 into parts >= 3.
%C Table 3.1: Partitions of integers congruent to 1 mod 3 into parts >= 3.
%C Table 3.2: Partitions of integers congruent to 2 mod 3 into parts >= 3.
%C And so on.
%C Conjecture:
%C Let j and n be integers congruent to k mod m such that 0 <= k < m <= j < n. Let h=(n-j)/m. Consider only all partitions of n into parts >= m. Then remove every partition in which the parts of size m appears a number of times < h. Then remove h parts of size m in every partition. The rest are the partitions of j into parts >= m. (Note that in the section model, h is the number of sections or subsections removed), (_Omar E. Pol_, Dec 05 2010, Dec 06 2010).
%C Starting from the first row of triangle, it appears that the total numbers of parts of size k in k successive rows give the sequence A000041 (see A182703). - _Omar E. Pol_, Feb 22 2012
%C The last section of n contains A187219(n) regions (see A206437). - _Omar E. Pol_, Nov 04 2012
%H Alois P. Heinz, <a href="/A135010/b135010.txt">Rows n = 1..23, flattened</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the section model of partitions</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the section model of partitions (2D view)</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the section model of partitions (3D view)</a>
%H Robert Price, <a href="/A135010/a135010.txt">Mathematica program to generate "Illustration of the section model of partitions"</a>
%H Robert Price, <a href="/A135010/a135010_1.txt">Mathematica program to generate "Illustration of the section model of partitions (2D view)"</a>
%e Triangle begins:
%e [1];
%e [1],[2];
%e [1],[1],[3];
%e [1],[1],[1],[2,2],[4];
%e [1],[1],[1],[1],[1],[2,3],[5];
%e [1],[1],[1],[1],[1],[1],[1],[2,2,2],[2,4],[3,3],[6];
%e ...
%e From _Omar E. Pol_, Sep 03 2013: (Start)
%e Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in the ordering mentioned in A026791. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
%e ---------------------------------------------------------
%e n j Diagram Parts Parts
%e ---------------------------------------------------------
%e . _
%e 1 1 |_| 1; 1;
%e . _
%e 2 1 | |_ 1, 1,
%e 2 2 |_ _| 2; 2;
%e . _
%e 3 1 | | 1, 1,
%e 3 2 | |_ _ 1, 1,
%e 3 3 |_ _ _| 3; 3;
%e . _
%e 4 1 | | 1, 1,
%e 4 2 | | 1, 1,
%e 4 3 | |_ _ _ 1, 1,
%e 4 4 | |_ _| 2,2, 2,2,
%e 4 5 |_ _ _ _| 4; 4;
%e . _
%e 5 1 | | 1, 1,
%e 5 2 | | 1, 1,
%e 5 3 | | 1, 1,
%e 5 4 | | 1, 1,
%e 5 5 | |_ _ _ _ 1, 1,
%e 5 6 | |_ _ _| 2,3, 2,3,
%e 5 7 |_ _ _ _ _| 5; 5;
%e . _
%e 6 1 | | 1, 1,
%e 6 2 | | 1, 1,
%e 6 3 | | 1, 1,
%e 6 4 | | 1, 1,
%e 6 5 | | 1, 1,
%e 6 6 | | 1, 1,
%e 6 7 | |_ _ _ _ _ 1, 1,
%e 6 8 | | |_ _| 2,2,2, 2,2,2,
%e 6 9 | |_ _ _ _| 2,4, 2,4,
%e 6 10 | |_ _ _| 3,3, 3,3,
%e 6 11 |_ _ _ _ _ _| 6; 6;
%e ...
%e (End)
%p with(combinat):
%p T:= proc(m) local b, ll;
%p b:= proc(n, i, l)
%p if n=0 then ll:=ll, l[]
%p else seq(b(n-j, j, [l[], j]), j=i..n)
%p fi
%p end;
%p ll:= NULL; b(m, 2, []); [1$numbpart(m-1)][], ll
%p end:
%p seq(T(n), n=1..10); # _Alois P. Heinz_, Feb 19 2012
%t less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[ Array[1 &, {PartitionsP[n - 1]}], Sort[ Reverse /@ Select[ IntegerPartitions[n], FreeQ[#, 1] &], less] ] // Flatten; Table[row[n], {n, 1, 9}] // Flatten (* _Jean-François Alcover_, Jan 14 2013 *)
%t Table[Reverse@ConstantArray[{1}, PartitionsP[n - 1]]~Join~
%t DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], {n, 1, 9}] // Flatten (* _Robert Price_, May 12 2020 *)
%Y Row n has length A138137(n).
%Y Row sums give A138879.
%Y Right border gives A000027.
%Y Cf. A000041, A026791, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207383, A207379, A211009.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Nov 17 2007, Mar 21 2008
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