A182709 -id:A182709 - cp's OEIS frontend

A198381 Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

0, 0, 0, 0, 1, 2, 6, 10, 20, 32, 54, 81, 128, 184, 273, 385, 549, 754, 1048, 1412, 1917, 2547, 3392, 4444, 5837, 7556, 9791, 12553, 16086, 20429, 25935, 32665, 41108, 51404, 64190, 79721, 98882, 122043, 150417, 184618, 226239

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Also partial sums of A182699. Total number of emergent parts in all partitions of the numbers <= n.

Also total number of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

Cf. A000041, A000065, A000070, A006128, A026905, A093694, A096541, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196930, A196931.

a(n) = A096541(n) - A000065(n) = 1 + A096541(n) - A000041(n) = 1 + A006128(n) - A000070(n).

a(n) = A006128(n) - A026905(n), n >= 1.

A220479 Total number of smallest parts that are also emergent parts in all partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 1, 5, 5, 10, 8, 22, 19, 33, 40, 62, 67, 107, 118, 175, 208, 282, 331, 462, 542, 712, 859, 1112, 1323, 1709, 2030, 2568, 3078, 3830, 4577, 5687, 6760, 8291, 9885, 12045, 14290, 17334, 20515, 24710, 29242, 35004, 41282, 49283, 57963, 68836

Offset: 1

Omar E. Pol, Jan 12 2013

nonn

For the definition of emergent parts see A182699.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Cf. A000041, A000070, A002865, A006128, A092269, A120452, A182699, A182709, A195820, A206437, A215513.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[b[n-i*j, i-1], {j, 0, n/i}]];
c[n_] := b[n, n];
d[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
a[n_] := c[n] - d[n+1];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

a(n) = A092269(n) - A000070(n-1) - A002865(n) = A092269(n) - A120452(n+1) = A195820(n) - A002865(n).
a(n) = A092269(n) - A000041(n) - A000070(n-2), n >= 2.
a(n) = A215513(n) - A000070(n-2), n >= 2.
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 31 2017

a(43) corrected by Vaclav Kotesovec, Jul 31 2017

A182708 a(n) is the sum of the smallest parts of all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 2, 3, 6, 7, 13, 14, 23, 27, 39, 45, 67, 75, 104, 125, 165, 194, 258, 302, 392, 467, 588, 700, 885, 1045, 1296, 1546, 1897, 2249, 2753, 3252, 3945, 4670, 5616, 6633, 7957, 9357, 11157, 13124, 15573, 18257, 21599, 25259, 29760, 34760, 40788, 47526, 55642, 64669, 75465, 87576, 101898, 117991, 136977, 158286

Offset: 1

Omar E. Pol, Nov 28 2010

In other words, sum of the smallest parts of all partitions of the head of the last section of the set of partitions of n.

Only one of the smallest parts is used in the sum.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A046746, A135010, A138121, A182707, A182709.

Table[Total[{Min /@ IntegerPartitions[n, All, Range[2, n]]}, 2], {n, 55}] (* Robert Price, Aug 30 2020 *) (* Only suitable for n<100 *)

my(N=66, z='z+O('z^N));  gf=sum(k=1, N, k * z^k / prod(j=k, N, 1-z^j ) ) - z/eta(z); concat([0], Vec(gf)) \\ Joerg Arndt, Aug 31 2020

a(n) = A046746(n) - A000041(n-1).

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (6*sqrt(2)*n^(3/2)) * (1 + (11*Pi/(24*sqrt(6)) - 3*sqrt(3/2)/Pi)/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 06 2019

A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 2, 2, 4, 2, 3, 6, 3, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 9, 4, 3, 3, 2, 2, 2, 2, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 4, 7, 3, 6, 5, 10, 5

Offset: 1

Omar E. Pol, Apr 18 2012

For the definition of "region of n" see A206437. See also A186114. Row n lists the largest part and the parts > 1 of the n-th region of the shell model of partitions. Also 1 together with the numbers > 1 of A206437.

			Written as a triangle begins:
1;
2;
3;
2;
4,2;
3;
5,2,
2;
4,2;
3;
6,3,2,2;
3;
5,2;
4;
7,3,2,2;

Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12

Column 1 is A141285. Records give A000027. The n-th record is T(A000041(n),1).

Cf. A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A198381, A206437, A210941.

A228109 Height after n-th step of an infinite staircase which is the lower part of a structure whose upper part is the infinite Dyck path of A228110.

Original entry on oeis.org

0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 1, 0, -1, 0, -1, 0, 1, 2, 1, 2, 3, 4, 3, 4, 3, 2, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 5, 4, 5, 4, 3, 2, 1, 0, -1

Offset: 0

Omar E. Pol, Aug 13 2013

sign

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 5, the diagram 1 represents the partitions of 5. The diagram 2 shows separately the boundary segments southwest sides of the first seven regions of the diagram 1, see below:
.
j   Diagram 1              Diagram 2
     _  
7 |  _    |                                |  _
6 |  _|  |                          | _       |
5 |     | |                  |                   |
4 | |_  | |              |     |_                |
3 |   | | |        |             |               |
2 |  | | | |    |      |           |               |
1 |||_|||  |  |    |          |              |_
.
.    1 2 3 4 5
.
a(n) is the height after n-th step of an infinite staircase which is the lower part of a diagram of regions of the set of partitions of all positive integers. The upper part of the diagram is the infinite Dyck path mentioned in A228110. The diagram shows the shape of the successive regions of the set of partitions of all positive integers. The area of the n-th region is A186412(n).
For the height of the peaks and the valleys in the infinite Dyck path see A229946.

			Illustration of initial terms (n = 1..53):
5
4                                                      /
3                                 /\/\                /
2                                /    \            /\/
1                   /\/\      /\/      \        /\/
0          /\    /\/    \    /          \    /\/
-1 \/\/\/\/  \/\/        \/\/            \/\/
-2
The diagram shows the Dyck pack mentioned in A228110 together with the staircase illustrated above. The area of the n-th region is equal to A186412(n).
.
7...................................
.                                  /\
5.....................            /  \                /\
.                    /\          /    \          /\  / /
3...........        /  \        / /\/\ \        /  \/ /
2......    /\      /    \    /\/ /    \ \      /   /\/
1...  /\  /  \  /\/ /\/\ \  / /\/      \ \  /\/ /\/
0  /\/  \/ /\ \/ /\/    \ \/ /          \ \/ /\/
-1 \/\/\/\/  \/\/        \/\/            \/\/
.
Region:
.   1  2    3   4     5      6      7       8    9   10

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228110, A229946.

A228110 Height after n-th step of the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, n >= 0, k >= 1.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Omar E. Pol, Aug 10 2013

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y].  See below:
.
.  j     Diagram 1         Partitions           Diagram 2
.        _   _                            _   _  
. 15  |  _       |      7                   _        |
. 14  |  _ |    |      4+3                  _ |    |
. 13  |  _    |   |      5+2                  _    |   |
. 12  |  _| |_  |      3+2+2                _| |_  |
. 11  |  _      | |      6+1                  _      | |
. 10  |  _|    | |      3+3+1               _ |    | |
.  9  |     |   | |      4+2+1                   |   | |
.  8  | |_ |  | |      2+2+2+1             |_ |  | |
.  7  |  _    | | |      5+1+1                _    | | |
.  6  |  _|  | | |      3+2+1+1             _ |  | | |
.  5  |     | | | |      4+1+1+1                 | | | |
.  4  | |_  | | | |      2+2+1+1+1           |_  | | | |
.  3  |   | | | | |      3+1+1+1+1             | | | | |
.  2  |  | | | | | |      2+1+1+1+1+1          | | | | | |
.  1  |||_|||_|_|      1+1+1+1+1+1+1       | | | | | | |
.
.      1 2 3 4 5 6 7
.
The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.
Also the diagram has the property that it can be transformed in a Dyck path (see example).
The sequence gives the height of the infinite Dyck path after n-th step.
The absolute values of the first differences give A000012.
For the height of the peaks and the valleys in the infinite Dyck path see A229946.
Q: Is this infinite Dyck path a fractal?

			Illustration of initial terms (n = 1..59):
.
11 ...........................................................
.                                                            /
.                                                           /
.                                                          /
7 ..................................                      /
.                                  /\                    /
5 ....................            /  \                /\/
.                    /\          /    \          /\  /
3 ..........        /  \        /      \        /  \/
2 .....    /\      /    \    /\/        \      /
1 ..  /\  /  \  /\/      \  /            \  /\/
.  /\/  \/    \/          \/              \/
.
Note that the j-th largest peak between two valleys at height 0 is also the partition number A000041(j).
Written as an irregular triangle in which row k has length 2*A138137(k), the sequence begins:
0,1;
0,1,2,1;
0,1,2,3,2,1;
0,1,2,1,2,3,4,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,4,3,2,1;
0,1,2,1,2,3,4,5,4,3,4,5,6,5,6,7,8,9,10,11,10,9,8,7,6,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,6,7,8,9,8,9,10,11,12,13,14,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1;
...

Omar E. Pol, Visualization of regions in a diagram for A006128

Column 1 is A000004. Both column 2 and the right border are in A000012. Both columns 3 and 5 are in A007395.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A229946.

A233968 Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.

Original entry on oeis.org

2, 4, 6, 12, 16, 30, 38, 64, 84, 128, 166, 248, 314, 448, 576, 790, 1004, 1358, 1708, 2264, 2844, 3694, 4614, 5936, 7354, 9342, 11544, 14502, 17816, 22220, 27144, 33584, 40878, 50192, 60828, 74276, 89596, 108778, 130772, 157918, 189116, 227374

Offset: 1

Omar E. Pol, Jan 14 2014

nonn

Also first differences of A211978.

			Illustration of initial terms as a dissection of a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                         _ _ _      |
.                                         _ _ _|_    |
.                                         _ _    |   |
.                             _ _ _ _ _      |   |   |
.                             _ _ _    |             |
.                   _ _ _ _        |   |             |
.                   _ _    |           |             |
.           _ _ _      |   |           |             |
.     _ _        |         |           |             |
. _      |       |         |           |             |
.  |     |       |         |           |             |
.
. 2    4      6       12          16          30
.
Also using the elements from the above diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
.  2, 4,   6,       12,           16,...
.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228109, A228110, A229946.

a(n) = 2*(A006128(n) - A006128(n-1)) = 2*A138137(n).

A182377 Total sum of positive ranks of all regions in the last shell of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 67, 91

Offset: 1

Omar E. Pol, Apr 29 2012

The rank of a region of n is the largest part minus the number of parts. For the definition of "region of n" see A206437. For the definition of "last shell of n" see A135010.

a(n) is also the sum of positive integers in row n of triangle A194447. First differs from A000094 at a(12).

			For n = 7 the last shell of 7 contains four regions: [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1] so we have:
----------------------------------------------------------
.        Largest    Number
Region     part    of parts    Rank
----------------------------------------------------------
.  1        3         1          2
.  2        5         2          3
.  3        4         1          3
.  4        7        15         -8
.
The sum of positive ranks is a(7) = 2 + 3 + 3 = 8.

Omar E. Pol, Illustration of the seven regions of 5

Cf. A000094, A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A206437, A209616.

A196039 Total sum of the smallest part of every partition of every shell of n.

Original entry on oeis.org

0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Partial sums of A046746.

Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012

			For n = 5 the seven partitions of 5 are:
5
3         + 2
4             + 1
2     + 2     + 1
3         + 1 + 1
2     + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
.
The five shells of 5 (see A135010 and also A138121), written as a triangle, are:
1
2, 1
3, 1, 1
4, (2, 2), 1, 1, 1
5, (3, 2), 1, 1, 1, 1, 1
.
The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
1
2, 1
3, 1, 1
4, 2, 1, 1, 1
5, 2, 1, 1, 1, 1, 1
.
thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Omar E. Pol, Illustration of the seven regions of 5

Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.

b:= proc(n, i) option remember;
     `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(nAlois P. Heinz, Apr 03 2012

b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

a(n) = A066186(n) - A196025(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*Pi*sqrt(2*n)). - Vaclav Kotesovec, Jul 06 2019

A220483 Total number of smallest parts that are also emergent parts in all partitions of n with at least one distinct part: a(n) = n + d(n) + p(n-1) + spt(n) - A000070(n) - sigma(n) - 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 8, 11, 19, 26, 34, 51, 67, 91, 118, 158, 200, 271, 331, 433, 538, 699, 849, 1089, 1323, 1674, 2030, 2542, 3066, 3813, 4567, 5640, 6760, 8272, 9871, 12002, 14290, 17287, 20515, 24675, 29214, 34981, 41282, 49216, 57957, 68798

Offset: 1

Omar E. Pol, Jan 16 2013

nonn

For the definition of "emergent part" see A182699, A182709.

Cf. A000005, A000041, A000070, A000203, A002865, A092269, A182699, A182709, A183152, A193827, A195820, A206437, A215513, A220479, A220489.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
a[n_] := n + DivisorSigma[0, n] + PartitionsP[n - 1] + b[n, n] -
  Total[PartitionsP[Range[0, n]]] - DivisorSigma[1, n] - 1;
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

a(n) = n + A000005(n) + A000041(n-1) + A092269(n) - A000070(n) - A000203(n) - 1.

a(49) corrected by Jean-François Alcover, Jun 05 2021

cp's OEIS Frontend

A198381 Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

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A220479 Total number of smallest parts that are also emergent parts in all partitions of n.

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A182708 a(n) is the sum of the smallest parts of all partitions of n that do not contain 1 as a part.

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A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

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A228109 Height after n-th step of an infinite staircase which is the lower part of a structure whose upper part is the infinite Dyck path of A228110.

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A228110 Height after n-th step of the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, n >= 0, k >= 1.

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A233968 Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.

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A182377 Total sum of positive ranks of all regions in the last shell of n.

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A196039 Total sum of the smallest part of every partition of every shell of n.

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