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%I #41 Jul 25 2020 12:00:45
%S 1,3,6,7,12,13,20,21,23,24,35,36,38,39,54,55,57,58,62,63,64,86,87,89,
%T 90,94,95,97,98,128,129,131,132,136,137,138,145,146,148,149,150,192,
%U 193,195,196,200,201,203,204,212,213,214,217,218,219,275
%N Total number of parts in the section model of partitions of A135010 with n regions.
%H Robert Price, <a href="/A182181/b182181.txt">Table of n, a(n) for n = 1..204226, rows 1-50.</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%F a(A000041(n)) = A006128(n), n >= 1.
%F a(A000041(n)) = A182727(A000041(n). - _Omar E. Pol_, May 24 2012
%e The first four regions of the section model of partitions are [1],[2, 1],[3, 1, 1],[2]. We can see that there are seven parts so a(4) = 7.
%e Written as a triangle begins:
%e 1;
%e 3;
%e 6;
%e 7, 12;
%e 13, 20;
%e 21, 23, 24, 35;
%e 36, 38, 39, 54;
%e 55, 57, 58, 62, 63, 64, 86;
%e 87, 89, 90, 94, 95, 97, 98,128;
%e 129,131,132,136,137,138,145,146,148,149,150,192;
%e 193,195,196,200,201,203,204,212,213,214,217,218,219,275;
%e ...
%e From _Omar E. Pol_, Oct 20 2014: (Start)
%e Illustration of initial terms:
%e . _ _ _ _ _
%e . _ _ _ |_ _ _ |
%e . _ _ _ _ |_ _ _|_ |_ _ _|_ |
%e . _ _ |_ _ | |_ _ | |_ _ | |
%e . _ _ _ |_ _|_ |_ _|_ | |_ _|_ | |_ _|_ | |
%e . _ _ |_ _ | |_ _ | |_ _ | | |_ _ | | |_ _ | | |
%e . _ |_ | |_ | | |_ | | |_ | | | |_ | | | |_ | | | |
%e . |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
%e .
%e . 1 3 6 7 12 13 20
%e .
%e . _ _ _ _ _ _
%e . _ _ _ |_ _ _ |
%e . _ _ _ _ |_ _ _|_ |_ _ _|_ |
%e . _ _ |_ _ | |_ _ | |_ _ | |
%e . |_ _|_ _ _ |_ _|_ _|_ |_ _|_ _|_ |_ _|_ _|_ |
%e . |_ _ _ | |_ _ _ | |_ _ _ | |_ _ _ | |
%e . |_ _ _|_ | |_ _ _|_ | |_ _ _|_ | |_ _ _|_ | |
%e . |_ _ | | |_ _ | | |_ _ | | |_ _ | | |
%e . |_ _|_ | | |_ _|_ | | |_ _|_ | | |_ _|_ | | |
%e . |_ _ | | | |_ _ | | | |_ _ | | | |_ _ | | | |
%e . |_ | | | | |_ | | | | |_ | | | | |_ | | | | |
%e . |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
%e .
%e . 21 23 24 35
%e (End)
%t lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
%t reg = {}; l = {};
%t For[j = 1, j <= 56, 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, j - i];
%t ];
%t Accumulate@reg (* _Robert Price_, Apr 22 2020, revised Jul 25 2020 *)
%Y Partial sums of A194446.
%Y Row j has length A187219(j).
%Y Right border gives A006128.
%Y For the definition of "region" see A206437.
%Y Cf. A000041, A135010, A138121, A141285, A182244, A182276, A182727, A186114, A210990.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Apr 23 2012
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