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%I
%S 1,2,1,3,1,1,4,2,2,1,1,1,5,3,2,1,1,1,1,1,6,3,3,4,2,2,2,2,1,1,1,1,1,1,
%T 1,7,4,3,5,2,3,2,2,1,1,1,1,1,1,1,1,1,1,1,8,4,4,5,3,6,2,3,3,2,4,2,2,2,
%U 2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,5,4,6,3,3,3,3,7,2,4,3,2,5,2,2,3,2,2
%N A shell model of partitions: Triangle read by rows in which row n lists the parts in the outer shell of the partitions of n.
%C Mirror of triangle A135010.
%H O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">A shell model of partitions (2D and 3D)</a> [From _Omar E. Pol_, Sep 07 2008]
%e Triangle begins:
%e 1
%e 2,1
%e 3,1,1
%e 4,2,2,1,1,1
%e 5,3,2,1,1,1,1,1,
%e 6,3,3,4,2,2,2,2,1,1,1,1,1,1,1
%e 7,4,3,5,2,3,2,2,1,1,1,1,1,1,1,1,1,1,1
%e ...
%e The illustration of the three views of the shell model of partitions, version "tree" with seven shells, shows the connection between several sequences.
%e ---------------------------------------------------------
%e Partitions A194805 Table 1.0
%e . of 7 p(n) A194551 A135010
%e ---------------------------------------------------------
%e 7 15 7 7 . . . . . .
%e 4+3 4 4 . . . 3 . .
%e 5+2 5 5 . . . . 2 .
%e 3+2+2 3 3 . . 2 . 2 .
%e 6+1 11 6 1 6 . . . . . 1
%e 3+3+1 3 1 3 . . 3 . . 1
%e 4+2+1 4 1 4 . . . 2 . 1
%e 2+2+2+1 2 1 2 . 2 . 2 . 1
%e 5+1+1 7 1 5 5 . . . . 1 1
%e 3+2+1+1 1 3 3 . . 2 . 1 1
%e 4+1+1+1 5 4 1 4 . . . 1 1 1
%e 2+2+1+1+1 2 1 2 . 2 . 1 1 1
%e 3+1+1+1+1 3 1 3 3 . . 1 1 1 1
%e 2+1+1+1+1+1 2 2 1 2 . 1 1 1 1 1
%e 1+1+1+1+1+1+1 1 1 1 1 1 1 1 1 1
%e . 1 ---------------
%e . *<------- A000041 -------> 1 1 2 3 5 7 11
%e . A182712 -------> 1 0 2 1 4 3
%e . A182713 -------> 1 0 1 2 2
%e . A182714 -------> 1 0 1 1
%e . 1 0 1
%e . A141285 A182703 1 0
%e . A182730 A182731 1
%e ---------------------------------------------------------
%e . A138137 --> 1 2 3 6 9 15..
%e ---------------------------------------------------------
%e . A182746 <--- 4 . 2 1 0 1 2 . 4 ---> A182747
%e ---------------------------------------------------------
%e .
%e . A182732 <--- 6 3 4 2 1 3 5 4 7 ---> A182733
%e . . . . . 1 . . . .
%e . . . . 2 1 . . . .
%e . . 3 . . 1 2 . . .
%e . Table 2.0 . . 2 2 1 . . 3 . Table 2.1
%e . . . . . 1 2 2 . .
%e . 1 . . . .
%e .
%e . A182982 A182742 A194803 A182983 A182743
%e . A182992 A182994 A194804 A182993 A182995
%e ---------------------------------------------------------
%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 // Reverse; Table[row[n], {n, 1, 9}] // Flatten (* _Jean-François Alcover_, Jan 15 2013 *)
%Y Mirror of triangle A135010. Row n has length A138137(n). Rows sums give A138879.
%Y Cf. A000041, A138879, A138880, A141285, A182703, A194812.
%K nonn,tabf,less
%O 1,2
%A _Omar E. Pol_, Mar 21 2008
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