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A182699 Number of emergent parts in all partitions of n.

%I A182699 #45 Jun 18 2015 08:51:53
%S A182699 0,0,0,0,1,1,4,4,10,12,22,27,47,56,89,112,164,205,294,364,505,630,845,
%T A182699 1052,1393,1719,2235,2762,3533,4343,5506,6730,8443,10296,12786,15531,
%U A182699 19161,23161,28374,34201,41621,49975,60513,72385,87200,103999,124670,148209
%N A182699 Number of emergent parts in all partitions of n.
%C A182699 Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n.
%C A182699 Also, here the "filler parts" of the partitions of n are defined to be the parts of the last section of the set of partitions of n that are not the emergent parts.
%C A182699 For n >= 4, length of row n of A183152. - _Omar E. Pol_, Aug 08 2011
%C A182699 Also total number of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (cf. A083751, A187219). - _Omar E. Pol_, Mar 04 2012
%H A182699 Alois P. Heinz, <a href="/A182699/b182699.txt">Table of n, a(n) for n = 0..1000</a>
%H A182699 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpanxt.jpg">Illustration: How to build the last section of the set of partitions (copy, paste and fill)</a>
%H A182699 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%F A182699 a(n) = A138135(n) - A002865(n), n >= 1.
%F A182699 From _Omar E. Pol_, Oct 21 2011: (Start)
%F A182699 a(n) = A006128(n) - A006128(n-1) - A000041(n), n >= 1.
%F A182699 a(n) = A138137(n) - A000041(n), n >= 1. (End)
%F A182699 a(n) = A076276(n) - A006128(n-1), n >= 1. - _Omar E. Pol_, Oct 30 2011
%e A182699 For n = 6 the partitions of 6 contain four "emergent" parts: (3), (4), (2), (2), so a(6) = 4. See below the location of the emergent parts.
%e A182699 6
%e A182699 (3) + 3
%e A182699 (4) + 2
%e A182699 (2) + (2) + 2
%e A182699 5 + 1
%e A182699 3 + 2 + 1
%e A182699 4 + 1 + 1
%e A182699 2 + 2 + 1 + 1
%e A182699 3 + 1 + 1 + 1
%e A182699 2 + 1 + 1 + 1 + 1
%e A182699 1 + 1 + 1 + 1 + 1 + 1
%e A182699 For a(10) = 22 see the link for the location of the 22 "emergent parts" (colored yellow and green) and the location of the 42 "filler parts" (colored blue) in the last section of the set of partitions of 10.
%p A182699 b:= proc(n, i) option remember; local t, h;
%p A182699       if n<0 then [0, 0, 0]
%p A182699     elif n=0 then [0, 1, 0]
%p A182699     elif i<2 then [0, 0, 0]
%p A182699     else t:= b(n, i-1); h:= b(n-i, i);
%p A182699          [t[1]+h[1]+h[2], t[2], t[3]+h[3]+h[1]]
%p A182699       fi
%p A182699     end:
%p A182699 a:= n-> b(n, n)[3]:
%p A182699 seq (a(n), n=0..50);  # _Alois P. Heinz_, Oct 21 2011
%t A182699 b[n_, i_] := b[n, i] = Module[{t, h}, Which[n<0, {0, 0, 0}, n == 0, {0, 1, 0}, i<2 , {0, 0, 0}, True, t = b[n, i-1]; h = b[n-i, i]; Join [t[[1]] + h[[1]] + h[[2]], t[[2]], t[[3]] + h[[3]] + h[[1]] ]]]; a[n_] := b[n, n][[3]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 18 2015, after _Alois P. Heinz_ *)
%Y A182699 Cf. A000041, A002865, A135010, A138121, A138135, A182700, A182709, A182740.
%K A182699 nonn,easy
%O A182699 0,7
%A A182699 _Omar E. Pol_, Nov 29 2010