A182699 Number of emergent parts in all partitions of n.

0, 0, 0, 0, 1, 1, 4, 4, 10, 12, 22, 27, 47, 56, 89, 112, 164, 205, 294, 364, 505, 630, 845, 1052, 1393, 1719, 2235, 2762, 3533, 4343, 5506, 6730, 8443, 10296, 12786, 15531, 19161, 23161, 28374, 34201, 41621, 49975, 60513, 72385, 87200, 103999, 124670, 148209

Offset: 0

Author

Omar E. Pol, Nov 29 2010

Keywords

• nonn
• easy

Comments

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.

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.

For n >= 4, length of row n of A183152. - Omar E. Pol, Aug 08 2011

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

Examples

			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.
6
(3) + 3
(4) + 2
(2) + (2) + 2
5 + 1
3 + 2 + 1
4 + 1 + 1
2 + 2 + 1 + 1
3 + 1 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Omar E. Pol, Illustration: How to build the last section of the set of partitions (copy, paste and fill)
• Omar E. Pol, Illustration of the shell model of partitions (2D view)

Crossrefs

Cf. A000041, A002865, A135010, A138121, A138135, A182700, A182709, A182740.

Programs

Maple

b:= proc(n, i) option remember; local t, h;
      if n<0 then [0, 0, 0]
    elif n=0 then [0, 1, 0]
    elif i<2 then [0, 0, 0]
    else t:= b(n, i-1); h:= b(n-i, i);
         [t[1]+h[1]+h[2], t[2], t[3]+h[3]+h[1]]
      fi
    end:
a:= n-> b(n, n)[3]:
seq (a(n), n=0..50);  # Alois P. Heinz, Oct 21 2011

Mathematica

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 *)

Formula

a(n) = A138135(n) - A002865(n), n >= 1.

From Omar E. Pol, Oct 21 2011: (Start)

a(n) = A006128(n) - A006128(n-1) - A000041(n), n >= 1.

a(n) = A138137(n) - A000041(n), n >= 1. (End)

a(n) = A076276(n) - A006128(n-1), n >= 1. - Omar E. Pol, Oct 30 2011