A138121 -id:A138121 - cp's OEIS frontend

A194439 Number of regions in the set of partitions of n that contain only one part.

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297

Offset: 1

Omar E. Pol, Nov 28 2011

It appears that this is 1 together with A000041. - Omar E. Pol, Nov 29 2011

For the definition of "region" see A206437. See also A186114 and A193870.

			For n = 5 the seven regions of 5 in nondecreasing order are the sets of positive integers of the rows as shown below:
   1;
   1, 2;
   1, 1, 3;
   0, 0, 0, 2;
   1, 1, 1, 2, 4;
   0, 0, 0, 0, 0, 3;
   1, 1, 1, 1, 1, 2, 5;
   ...
There are three regions that contain only one positive part, so a(5) = 3.
Note that in every column of the triangle the positive integers are also the parts of one of the partitions of 5.

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

Column 1 of A194438.

Cf. A000041, A002865, A027336, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194446, A194447, A206437.

It appears that a(n) = A000041(n-2), if n >= 2. - Omar E. Pol, Nov 29 2011

It appears that a(n) = A000041(n) - A027336(n), if n >= 2. - Omar E. Pol, Nov 30 2011

Definition clarified by Omar E. Pol, May 21 2021

A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 0, 3, 3, 4, 0, 4, 5, 2, 3, 0, 5, 7, 8, 6, 4, 0, 6, 11, 6, 6, 4, 5, 0, 7, 15, 16, 9, 12, 5, 6, 0, 8, 22, 14, 18, 8, 10, 6, 7, 0, 9, 30, 30, 18, 20, 15, 12, 7, 8, 0, 10, 42, 30, 30, 20, 20, 12, 14, 8, 9, 0, 11, 56, 54, 42, 40, 25, 30, 14, 16, 9, 10, 0, 12

Offset: 1

Omar E. Pol, Feb 24 2012

For further properties of this triangle see also A182703.

			Triangle begins:
   1;
   1,  2;
   2,  0,  3;
   3,  4,  0,  4;
   5,  2,  3,  0,  5;
   7,  8,  6,  4,  0,  6;
  11,  6,  6,  4,  5,  0,  7;
  15, 16,  9, 12,  5,  6,  0,  8;
  22, 14, 18,  8, 10,  6,  7,  0,  9;
  30, 30, 18, 20, 15, 12,  7,  8,  0, 10;
  42, 30, 30, 20, 20, 12, 14,  8,  9,  0, 11;
  56, 54, 42, 40, 25, 30, 14, 16,  9, 10,  0, 12;
...
From _Omar E. Pol_, Nov 28 2020: (Start)
Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
.                                        _ _ _ _ _ _ _
.     (7)                    (7)        |_ _ _ _      |
.     (4+3)                (4+3)        |_ _ _ _|_    |
.     (5+2)                (5+2)        |_ _ _    |   |
.     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
.       (1)                  (1)                    | |
.         (1)                (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.           (1)              (1)                    | |
.             (1)            (1)                    | |
.             (1)            (1)                    | |
.               (1)          (1)                    | |
.                 (1)        (1)                    |_|
.    ----------------
.     19,8,5,3,2,1,1 --> Row 7 of triangle A207031
.      |/|/|/|/|/|/|
.     11,3,2,1,1,0,1 --> Row 7 of triangle A182703
.      * * * * * * *
.      1,2,3,4,5,6,7 --> Row 7 of triangle A002260
.      = = = = = = =
.     11,6,6,4,5,0,7 --> Row 7 of this triangle
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So the 7th row of triangle is [11, 6, 6, 4, 5, 0, 7]. (End)

Alois P. Heinz, Rows n = 1..141, flattened

Column 1 is A000041.
Leading diagonal gives A000027.
Second diagonal gives A000007.
Row sums give A138879.
Cf. A002260, A066186, A135010, A138121, A138785, A138880, A182703, A194812, A207031.

T(n,k) = k*A182703(n,k).

A183152 Irregular triangle read by rows in which row n lists the emergent parts of all partitions of n, or 0 if such parts do not exist.

Original entry on oeis.org

0, 0, 0, 0, 2, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 4, 7, 3, 6, 5, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 9, 4, 3, 3, 2, 2, 2, 2, 5, 4, 8, 4, 3, 7, 6

Offset: 0

Omar E. Pol, Aug 07 2011

For the definition of "emergent part" see A182699 and also A182709.

Also [0, 0, 0, 0] followed by the positive integers of the rows that contain zeros in the triangle A193870. For another version see A193827. - Omar E. Pol, Aug 12 2011

			If written as a triangle:
0,
0,
0,
0,
2,
3,
2,4,2,3,
3,5,2,4,
2,4,2,3,6,3,2,2,5,4,
3,5,2,4,7,3,2,2,3,6,3,5,
2,4,2,3,6,3,2,2,5,4,8,4,3,2,2,2,2,4,7,3,6,5,
3,5,2,4,7,3,2,2,3,6,3,5,9,4,3,3,2,2,2,2,5,4,8,4,3,7,6

Row n has length A182699(n). Row sums give A182709.

Cf. A135010, A138121, A182707.

A182746 Bisection (even part) of number of partitions that do not contain 1 as a part A002865.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 88, 137, 210, 320, 478, 708, 1039, 1507, 2167, 3094, 4378, 6153, 8591, 11914, 16424, 22519, 30701, 41646, 56224, 75547, 101066, 134647, 178651, 236131, 310962, 408046, 533623, 695578, 903811, 1170827, 1512301, 1947826, 2501928

Offset: 0

Omar E. Pol, Dec 01 2010

a(n+1) is the number of partitions p of 2n-1 such that (number of parts of p) is a part of p, for n >=0. - Clark Kimberling, Mar 02 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, and Gideon Vos, Large-N correlation functions in N = 2 superconformal QCD, arXiv preprint arXiv:1610.07612 [hep-th], 2016.
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.

Cf. A000041, A002865, A058696, A135010, A138121, A182740, A182742, A182743, A182747.

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
a:= n-> b(2*n, 2*n):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

Table[Count[IntegerPartitions[2 n -1], p_ /; MemberQ[p, Length[p]]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n, 2*n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
a[n_] := PartitionsP[2*n] - PartitionsP[2*n - 1]; Table[a[n], {n, 0, 40}] (* George Beck, Jun 05 2017 *)

a(n)=numbpart(2*n)-numbpart(2*n-1) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = p(2*n) - p(2*n-1), where p is the partition function, A000041. - George Beck, Jun 05 2017 [Shifted by Georg Fischer, Jun 20 2022]

More terms from Alois P. Heinz, Dec 01 2010

A182747 Bisection (odd part) of number of partitions that do not contain 1 as a part A002865.

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 24, 41, 66, 105, 165, 253, 383, 574, 847, 1238, 1794, 2573, 3660, 5170, 7245, 10087, 13959, 19196, 26252, 35717, 48342, 65121, 87331, 116600, 155038, 205343, 270928, 356169, 466610, 609237, 792906, 1028764, 1330772, 1716486, 2207851

Offset: 0

Omar E. Pol, Dec 01 2010

a(n+1) = number of partitions p of 2n such that (number of parts of p) is a part of p, for n >=0. - Clark Kimberling, Mar 02 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A002865, A058695, A135010, A138121, A182740, A182742, A182743, A182746.

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
a:= n-> b(2*n+1, 2*n+1):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

f[n_] := Table[PartitionsP[2 k + 1] - PartitionsP[2 k], {k, 0, n}] (* George Beck, Aug 14 2011 *)
(* also *)
Table[Count[IntegerPartitions[2 n], p_ /; MemberQ[p, Length[p]]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n == 0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n+1, 2*n+1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = p(2*n+1)-p(2*n), where p is the partition function, A000041. - George Beck, Aug 14 2011

More terms from Alois P. Heinz, Dec 01 2010

A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 5, 12, 13, 22, 22, 43, 43, 67, 81, 117, 133, 195, 223, 312, 373, 492, 584, 782, 925, 1190, 1433, 1820, 2170, 2748, 3268, 4075, 4872, 5997, 7150, 8781, 10420, 12669, 15055, 18198, 21535, 25925, 30602, 36624, 43201, 51428, 60478, 71802, 84215

Offset: 1

Omar E. Pol, Oct 19 2011

nonn

Total number of smallest parts in all partitions of the head of the last section of the set of partitions of n.

			For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
.  (8)
.  (4) + (4)
.   5  + (3)
.   6  + (2)
.   3  +  3  + (2)
.   4  + (2) + (2)
.  (2) + (2) + (2) + (2)
Note that in every partition the smallest parts are shown between parentheses. The total number of smallest parts is 1+2+1+1+1+2+4 = 12, so a(8) = 12.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, The number of smallest parts in the partitions of n
A. Folsom and K. Ono, The spt-function of Andrews
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2020.
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13, arXiv:1011.1955 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function
Wikipedia, Spt function

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984.

b:= proc(n, i) option remember;
      `if`(n=0 or i<2, 0, b(n, i-1)+
       add(`if`(n=i*j, j, b(n-i*j, i-1)), j=1..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..60); # Alois P. Heinz, Apr 09 2012

Table[s = Select[IntegerPartitions[n], ! MemberQ[#, 1] &]; Plus @@ Table[Count[x, Min[x]], {x, s}], {n, 50}] (* T. D. Noe, Oct 19 2011 *)
b[n_, i_] := b[n, i] = If[n==0 || i<2, 0, b[n, i-1] + Sum[If[n== i*j, j, b[n-i*j, i-1]], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

def A195820(n):
    return sum(list(p).count(min(p)) for p in Partitions(n,min_part=2))
# D. S. McNeil, Oct 19 2011

a(n) = A092269(n) - A000070(n-1).
G.f.: Sum_{i>=2} x^i/(1 - x^i) * Product_{j>=i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 - (72 + 5*Pi^2)*sqrt(6) / (144*Pi*sqrt(n))). - Vaclav Kotesovec, Jul 31 2017

More terms from D. S. McNeil, Oct 19 2011

A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 3, 0, 1, 7, 1, 2, 0, 1, 9, 6, 2, 2, 0, 1, 15, 4, 4, 1, 2, 0, 1, 19, 13, 4, 5, 1, 2, 0, 1, 32, 10, 10, 3, 4, 1, 2, 0, 1, 40, 24, 10, 9, 4, 4, 1, 2, 0, 1, 60, 23, 18, 8, 8, 3, 4, 1, 2, 0, 1, 78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1

Offset: 1

Omar E. Pol, Feb 17 2012

For the calculation of row n, the number of odd/even parts, etc, take the row n from the triangle A207031 and then follow the same rules of A206563.

			Triangle begins:
  1;
  1,   1;
  3,   0,  1;
  3,   3,  0,  1;
  7,   1,  2,  0, 1;
  9,   6,  2,  2, 0, 1;
  15,  4,  4,  1, 2, 0, 1;
  19, 13,  4,  5, 1, 2, 0, 1;
  32, 10, 10,  3, 4, 1, 2, 0, 1;
  40, 24, 10,  9, 4, 4, 1, 2, 0, 1;
  60, 23, 18,  8, 8, 3, 4, 1, 2, 0, 1;
  78, 46, 22, 19, 8, 9, 3, 4, 1, 2, 0, 1;

Cf. A006128, A066897, A066898, A135010, A138121, A138135, A138137, A141285, A181187, A182703, A206563, A207031.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A207031(n,j)).

It appears that A182703(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 1, 2, 2, 2, 1, 1, 4, 3, 2, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 4, 4, 2, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 29 2020

T(n,k) is also the number of divisors of A336811(n,k).

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

			Triangle begins:
  1;
  2;
  2, 1;
  3, 2, 1;
  2, 2, 2, 1, 1;
  4, 3, 2, 2, 2, 1, 1;
  2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1;
  4, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1;
  3, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of the triangle, flattened)

Number of divisors of A336811.
Row n has length A000041(n-1).
Every column gives A000005.
Row sums give A138137 (conjectured).
Cf. A135010, A138121, A138879, A187219.

A339304row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]];Array[A339304row,10] (* Paolo Xausa, Sep 01 2023 *)

a(m) = A000005(A336811(m)).

T(n,k) = A000005(A336811(n,k)).

A182732 The limit of row A182730(n,.) as n-> infinity.

Original entry on oeis.org

2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 4, 8, 7, 6, 12, 6, 11, 10, 9, 8, 16, 3, 6, 5, 9, 4, 8, 7, 6, 12, 7, 6, 11, 5, 10, 9, 8, 15, 6, 5, 10, 9, 8, 7, 14, 8, 7, 13, 6, 12, 11, 10, 9, 18

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Largest part of the n-th partition of the table 2.0 mentioned in A135010. For the table 2.0 see A182982.

Omar E. Pol, Illustration of the shell model of partitions (3D view)

One together with where records occur give A182746.

Cf. A135010, A138121, A141285, A151285, A182730, A182731, A182733, A182982, A182983.

A182733 The limit of row A182731(n,.) as n-> infinity.

Original entry on oeis.org

3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 3, 6, 5, 9, 4, 8, 7, 6, 12, 7, 6, 11, 5, 10, 9, 8, 15, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 5, 9, 8, 7, 13, 7, 6, 12, 11, 10, 9, 17, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 4, 8, 7, 6, 12, 6, 11, 10, 9, 8, 16, 7, 6, 11, 5, 10, 9, 8, 15, 9, 8, 7, 14, 7, 13, 12, 11, 10, 19

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Largest part of the n-th partition of the table 2.1 mentioned in A135010. For the table 2.1 see A182983.

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Zero together with where records occur give A182747.

Cf. A135010, A138121, A141285, A182730, A183731, A182732, A182982, A182983.

cp's OEIS Frontend

A194439 Number of regions in the set of partitions of n that contain only one part.

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A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.

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A183152 Irregular triangle read by rows in which row n lists the emergent parts of all partitions of n, or 0 if such parts do not exist.

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A182746 Bisection (even part) of number of partitions that do not contain 1 as a part A002865.

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A182747 Bisection (odd part) of number of partitions that do not contain 1 as a part A002865.

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A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

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A207032 Triangle read by rows: T(n,k) = number of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

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A339304 Irregular triangle read by rows T(n,k) in which row n has length the partition number A000041(n-1) and columns k give the number of divisors function A000005, 1 <= k <= n.

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