A026794 -id:A026794 - cp's OEIS frontend

A026807 Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n.

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 2, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 4, 2, 1, 1, 1, 1, 22, 7, 3, 2, 1, 1, 1, 1, 30, 8, 4, 2, 1, 1, 1, 1, 1, 42, 12, 5, 3, 2, 1, 1, 1, 1, 1, 56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1, 77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1, 101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 135, 34, 13

Offset: 1

Clark Kimberling

T(n,g) is also the number of not necessarily connected 2-regular graphs with girth at least g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012

			Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))) = y*x+(2*y+y^2)*x^2+(3*y+y^2+y^3)*x^3+(5*y+2*y^2+y^3+y^4)*x^4+(7*y+2*y^2+y^3+y^4+y^5)*x^5+...
Triangle starts:  - _Jason Kimberley_, Feb 05 2012
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
7, 2, 1, 1, 1;
11, 4, 2, 1, 1, 1;
15, 4, 2, 1, 1, 1, 1;
22, 7, 3, 2, 1, 1, 1, 1;
30, 8, 4, 2, 1, 1, 1, 1, 1;
42, 12, 5, 3, 2, 1, 1, 1, 1, 1;
56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1;
77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1;
101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
From _Tilman Piesk_, Feb 20 2016: (Start)
n = 12, k = 4, t = A000217(k-1) = 6
vp = A000041(n..n-t) = A000041(12..6) = (77, 56, 42, 30, 22, 15, 11)
vc = A231599(k-1, 0..t) = A231599(3, 0..6) = (1,-1,-1, 0, 1, 1,-1)
T(12, 4) = vp * transpose(vc) = 77-56-42+22+15-11 = 5
(End)

Alois P. Heinz, Rows n = 1..141, flattened
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Tilman Piesk, Table for n = 1..30, table for n = 2..150 without values 1, illustrations of columns n = 2, 3, 4, 5, 6, 7, 8

Row sums give A046746.
Cf. A026835.
Cf. A026794.
Cf. A231599.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: this sequence (triangle); columns of this sequence: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular simple graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012
Cf. A002260.

import Data.List (tails)
a026807 n k = a026807_tabl !! (n-1) !! (k-1)
a026807_row n = a026807_tabl !! (n-1)
a026807_tabl = map
   (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
   where p 0  _ = 1
         p _ [] = 0
         p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 01 2012

T:= proc(n, k) option remember;
      `if`(k<1 or k>n, 0, `if`(n=k, 1, T(n, k+1) +T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Mar 28 2012

T[n_, k_] := T[n, k] = If[ k<1 || k>n, 0, If[n == k, 1, T[n, k+1] + T[n-k, k]]]; Table [Table[ T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

from see_there import a231599_row  # A231599
from sympy.ntheory import npartitions  # A000041
def a026807(n, k):
    if k > n:
        return 0
    elif k > n/2:
        return 1
    else:
        vc = a231599_row(k-1)
        t = len(vc)
        vp_range = range(n-t, n+1)
        vp_range = vp_range[::-1]  # reverse
        r = 0
        for i in range(0, t):
            r += vc[i] * npartitions(vp_range[i])
        return r
# Tilman Piesk, Feb 21 2016

T(n,1)=A000041(n), T(n,2)=A002865(n) for n>1, T(n,3)=A008483(n) for n>2, T(n,4)=A008484(n) for n>3.
G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic, Jun 22 2003
T(n, k) = T(n, k+1) + T(n-k, k), T(n, k) = 1 if n/2 < k <= n. - Franklin T. Adams-Watters, Jan 24 2005; Tilman Piesk, Feb 20 2016
T(n, k) = A000041(n..n-t) * transpose(A231599(k-1, 0..t)) with t = A000217(k-1). - Tilman Piesk, Feb 20 2016
Equals A026794 * A000012 as infinite lower triangular matrices. - Gary W. Adamson, Jan 31 2008

A185325 Number of partitions of n into parts >= 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 13, 15, 18, 21, 26, 30, 36, 42, 50, 58, 70, 80, 95, 110, 129, 150, 176, 202, 236, 272, 317, 364, 423, 484, 560, 643, 740, 847, 975, 1112, 1277, 1456, 1666, 1897, 2168, 2464, 2809, 3189, 3627, 4112, 4673

Offset: 0

Jason Kimberley, Nov 11 2011

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 5 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.
By removing a single part of size 5, an A026798 partition of n becomes an A185325 partition of n - 5. Hence this sequence is essentially the same as A026798.
a(n) = number of partitions of n+4 such that 4*(number of parts) is a part. - Clark Kimberling, Feb 27 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

2-regular simple graphs with girth at least 5: A185115 (connected), A185225 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), this sequence (g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: this sequence (k=2), A185335 (k=3).
Cf. A008484, A008483.

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A185325 := func;
[A185325(n):n in[0..60]];

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+5): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(1/mul(1-x^(m+5), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Length[p]]], {n, 40}], 3]  (* Clark Kimberling, Feb 27 2014 *)
CoefficientList[Series[1/QPochhammer[x^5, x], {x, 0, 70}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+5))) \\ G. C. Greubel, Nov 03 2019

def A185325_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+5)) for m in (0..80)) ).list()
A185325_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=5} 1/(1-x^m).
Given by p(n) -p(n-1) -p(n-2) +2*p(n-5) -p(n-8) -p(n-9) +p(n-10), where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010 [sign of 10 corrected from + to -, and moved from A026798 to this sequence by Jason Kimberley].
This sequence is the Euler transformation of A185115.
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (6*sqrt(3)*n^3). - Vaclav Kotesovec, Jun 02 2018
G.f.: exp(Sum_{k>=1} x^(5*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018
G.f.: 1 + Sum_{n >= 1} x^(n+4)/Product_{k = 0..n-1} (1 - x^(k+5)). - Peter Bala, Dec 01 2024

A124943 Table read by rows: number of partitions of n with k as low median.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1, 6, 3, 1, 0, 0, 1, 8, 4, 2, 0, 0, 0, 1, 11, 6, 3, 1, 0, 0, 0, 1, 15, 8, 4, 2, 0, 0, 0, 0, 1, 20, 12, 5, 3, 1, 0, 0, 0, 0, 1, 26, 16, 7, 4, 2, 0, 0, 0, 0, 0, 1, 35, 22, 10, 5, 3, 1, 0, 0, 0, 0, 0, 1, 45, 29, 14, 6, 4, 2, 0, 0, 0, 0, 0, 0, 1, 58, 40, 19, 8, 5, 3, 1

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

For a multiset with an odd number of elements, the low median is the same as the median. For a multiset with an even number of elements, the low median is the smaller of the two central elements.

Arrange the parts of a partition nonincreasing order. Remove the first part, then the last, then the first remaining part, then the last remaining part, and continue until only a single number, the low median, remains. - Clark Kimberling, May 16 2019

			For the partition [2,1^2], the sole middle element is 1, so that is the low median. For [3,2,1^2], the two middle elements are 1 and 2; the low median is the smaller, 1.
First 8 rows:
  1
  1   1
  2   0   1
  3   1   0   1
  4   2   0   0   1
  6   3   1   0   0   1
  8   4   2   0   0   0   1
  11  6   3   1   0   0   0   1
From _Gus Wiseman_, Jul 09 2023: (Start)
Row n = 8 counts the following partitions:
  (71)        (62)     (53)   (44)  .  .  .  (8)
  (611)       (521)    (431)
  (5111)      (422)    (332)
  (4211)      (3221)
  (41111)     (2222)
  (3311)      (22211)
  (32111)
  (311111)
  (221111)
  (2111111)
  (11111111)
(End)

Row sums are A000041.
Column k = 1 is A027336, ranks A363488.
The high version of this triangle is A124944.
The rank statistic for this triangle is A363941, high version A363942.
A version for mean instead of median is A363945, rank statistic A363943.
A high version for mean instead of median is A363946, rank stat A363944.
A version for mode instead of median is A363952, high A363953.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A325347 counts partitions with integer median, ranks A359908.
A359893 and A359901 count partitions by median.
A360005(n)/2 returns median of prime indices.
Cf. A025065, A026794, A027193, A067538, A237984, A240219, A362608, A363740.

Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 2)/2]]] &, IntegerPartitions[#]]] &, Range[13]]  (* Peter J. C. Moses, May 14 2019 *)

A026797 Number of partitions of n in which the least part is 4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 24, 27, 34, 39, 50, 57, 70, 81, 100, 115, 140, 161, 195, 225, 269, 311, 371, 427, 505, 583, 688, 791, 928, 1067, 1248, 1434, 1668, 1914, 2223, 2546, 2945, 3370, 3889

Offset: 1

Clark Kimberling

a(n) is also the number of, not necessarily connected, 2-regular simple graphs girth exactly 4. - Jason Kimberley, Feb 22 2013

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same as A008484.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), this sequence (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), A185644 (triangle); fixed k: this sequence (k=2), A185134 (k=3), A185144 (k=4).

R:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^4/(&*[1-x^(m+4): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^4/mul(1-x^(m+4), m=0..65), x, n+1), x, n), n = 1..60); # G. C. Greubel, Nov 03 2019

Table[Count[IntegerPartitions[n],?(Min[#]==4&)],{n,60}] (* _Harvey P. Dale, May 13 2012 *)
Rest@CoefficientList[Series[x^4/QPochhammer[x^4, x], {x,0,60}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^60)); concat([0,0,0], Vec(x^4/prod(m=0,70, 1-x^(m+4)))) \\ G. C. Greubel, Nov 03 2019

def A026797_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^4/product((1-x^(m+4)) for m in (0..60)) ).list()
a=A026797_list(60); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^4 * Product_{m>=4} 1/(1-x^m).
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (12*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(4*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

A124944 Table, number of partitions of n with k as high median.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 4, 1, 1, 1, 1, 1, 8, 6, 3, 1, 1, 1, 1, 1, 11, 8, 5, 1, 1, 1, 1, 1, 1, 15, 11, 7, 3, 1, 1, 1, 1, 1, 1, 20, 15, 9, 5, 1, 1, 1, 1, 1, 1, 1, 26, 21, 12, 8, 3, 1, 1, 1, 1, 1, 1, 1, 35, 27, 16, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 45, 37, 21, 13, 8, 3

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - Peter Munn, Jul 16 2017
Arrange the parts of a partition nonincreasing order.  Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - Clark Kimberling, May 14 2019

			For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
From _Gus Wiseman_, Jul 12 2023: (Start)
Triangle begins:
   1
   1  1
   1  1  1
   2  1  1  1
   3  1  1  1  1
   4  3  1  1  1  1
   6  4  1  1  1  1  1
   8  6  3  1  1  1  1  1
  11  8  5  1  1  1  1  1  1
  15 11  7  3  1  1  1  1  1  1
  20 15  9  5  1  1  1  1  1  1  1
  26 21 12  8  3  1  1  1  1  1  1  1
  35 27 16 10  5  1  1  1  1  1  1  1  1
  45 37 21 13  8  3  1  1  1  1  1  1  1  1
  58 48 29 16 11  5  1  1  1  1  1  1  1  1  1
Row n = 8 counts the following partitions:
  (611)       (521)    (431)   (44)  (53)  (62)  (71)  (8)
  (5111)      (422)    (332)
  (41111)     (4211)   (3311)
  (32111)     (3221)
  (311111)    (2222)
  (221111)    (22211)
  (2111111)
  (11111111)
(End)

Row sums are A000041.
Column k = 1 is A027336(n-1), ranks A364056.
Column k = 1 in the low version is A027336, ranks A363488.
The low version of this triangle is A124943.
The rank statistic for this triangle is A363942, low version A363941.
A version for mean instead of median is A363946, low A363945.
A version for mode instead of median is A363953, low A363952.
A008284 counts partitions by length, maximum, or decreasing mean.
A026794 counts partitions by minimum, strict A026821.
A325347 counts partitions with integer median, ranks A359908.
A359893 and A359901 count partitions by median.
A360005(n)/2 returns median of prime indices.
Cf. A008289, A025065, A027193, A067538, A237984, A240219, A362608, A363740, A363943, A363944.

Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 1)/2]]] &, IntegerPartitions[#]]] &, Range[13]]  (* Peter J. C. Moses, May 14 2019 *)

A185326 Number of partitions of n into parts >= 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151

Offset: 0

Jason Kimberley, Jan 30 2012

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 6 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

By removing a single part of size 6, an A026799 partition of n becomes an A185326 partition of n - 6. Hence this sequence is essentially the same as A026799.

Jason Kimberley, Table of n, a(n) for n = 0..998
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

2-regular simple graphs with girth at least 6: A185116 (connected), A185226 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), this sequence (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

Magma
```
A185326 := func;
```

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+6): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(1/mul(1-x^(m+6), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

CoefficientList[Series[1/QPochhammer[x^6, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+6))) \\ G. C. Greubel, Nov 03 2019

def A185326_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+6)) for m in (0..80)) ).list()
A185326_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=6} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-6) + p(n-7) - p(n-8) - p(n-9) - p(n-10) + p(n-13) + p(n-14) - p(n-15) where p(n) = A000041(n).
a(n) = A185226(n) + A185116(n).
This sequence is the Euler transformation of A185116.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(6*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
G.f.: 1 + Sum_{n >= 1} x^(n+5)/Product_{k = 0..n-1} (1 - x^(k+6)). - Peter Bala, Dec 01 2024

A026798 Number of partitions of n in which the least part is 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 13, 15, 18, 21, 26, 30, 36, 42, 50, 58, 70, 80, 95, 110, 129, 150, 176, 202, 236, 272, 317, 364, 423, 484, 560, 643, 740, 847, 975, 1112, 1277, 1456, 1666, 1897, 2168

Offset: 0

Clark Kimberling

Also the number of not necessarily connected 2-regular simple graphs with girth exactly 5. - Jason Kimberley, Nov 11 2011

Such partitions of n+5 correspond to A185325 partitions (parts >= 5) of n by removing a single part of size 5. - Jason Kimberley, Nov 11 2011

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same as A185325.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), this sequence (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Nov 11 2011

R:=PowerSeriesRing(Integers(), 60); [1,0,0,0,0] cat Coefficients(R!( x^5/(&*[1-x^(m+5): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

ZL := [ B,{B=Set(Set(Z, card>=5))}, unlabeled ]: 1,0,0,0,0, seq(combstruct[count](ZL, size=n), n=0..54); # Zerinvary Lajos, Mar 13 2007
1, seq(coeff(series(x^5/mul(1-x^(m+5), m=0..70), x, n+1), x, n), n = 0..65); # G. C. Greubel, Nov 03 2019

f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Join[{1, 0, 0, 0, 0, 1}, Table[ f[n, 5], {n, 50}]] (* Robert G. Wilson v *)
Join[{1}, Drop[CoefficientList[Series[x^5/QPochhammer[x^5, x], {x,0,60}], x], 1]] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^60)); concat([1,0,0,0,0], Vec(x^5/prod(m=0,70, 1-x^(m+5)))) \\ G. C. Greubel, Nov 03 2019

def A026798_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^5/product((1-x^(m+5)) for m in (0..70)) ).list()
a=A026798_list(65); [1]+a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^5 * Product_{m>=5} 1/(1-x^m).
a(n+5) is given by p(n) - p(n-1) - p(n-2) + 2p(n-5) - p(n-8) - p(n-9) + p(n-10) where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010 [sign of 10 and offset of formula corrected by Jason Kimberley, Nov 11 2011]
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (6*sqrt(3)*n^3). - Vaclav Kotesovec, Jun 02 2018

A185329 Number of partitions of n with parts >= 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 24, 26, 30, 34, 39, 43, 50, 55, 63, 71, 80, 89, 102, 113, 128, 143, 161, 179, 203, 225, 253, 282, 316, 351, 395, 437, 489, 544, 607, 673, 752, 832, 927, 1028, 1143

Offset: 0

Jason Kimberley, Feb 01 2012

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 9 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.
By removing a single part of size 9, an A026802 partition of n becomes an A185329 partition of n - 9. Hence this sequence is essentially the same as A026802.
In general, if g>=1 and g.f. = Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, Jun 02 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), this sequence (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+9): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(1/mul(1-x^(m+9), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

CoefficientList[Series[x^9/QPochhammer[x^9, x], {x,0,75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+9))) \\ G. C. Greubel, Nov 03 2019

def A185329_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+9)) for m in (0..80)) ).list()
A185329_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=9} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-9) - p(n-11) - 2*p(n-12) - p(n-13) - p(n-15) + p(n-16) + p(n-17) + 2*p(n-18) + p(n-19) + p(n-20) - p(n-21) - p(n-23) - 2*p(n-24) - p(n-25) + p(n-27) + p(n-29) + p(n-31) - p(n-34) - p(n-35) + p(n-36) where p(n)=A000041(n). - Shanzhen Gao
This sequence is the Euler transformation of A185119.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(9*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
G.f.: 1 + Sum_{n >= 1} x^(n+8)/Product_{k = 0..n-1} (1 - x^(k+9)). - Peter Bala, Dec 01 2024

A026799 Number of partitions of n in which the least part is 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151

Offset: 0

Clark Kimberling

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 6 (all such graphs are simple). Each integer part i corresponds to an i-cycle; the addition of integers corresponds to the disconnected union of cycles.

			a(0)=0 because there does not exist a least part of the empty partition.
The  a(6)=1 partition is 6.
The a(12)=1 partition is 6+6.
The a(13)=1 partition is 6+7.
.............................
The a(17)=1 partition is 6+11.
The a(18)=2 partitions are 6+6+6 and 6+12.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same as A185326.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), this sequence (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 04 2011

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026799 := func< n | p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)- p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) >; // Jason Kimberley, Feb 04 2011

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0] cat Coefficients(R!( x^6/(&*[1-x^(m+6): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

ZL := [ B,{B=Set(Set(Z, card>=6))}, unlabeled ]: 0,0,0,0,0,0, seq(combstruct[count](ZL, size=n), n=0..63); # Zerinvary Lajos, Mar 13 2007
seq(coeff(series(x^6/mul(1-x^(m+6), m=0..70), x, n+1), x, n), n = 0..65); # G. C. Greubel, Nov 03 2019

f[1, 1]=f[0, k_]=1; f[n_, k_]:= f[n, k] = If[n<0, 0, If[k>n, 0, If[k==n, 1, f[n, k+1] +f[n-k, k]]]]; Join[{0,0,0,0,0,0}, Table[f[n, 6], {n, 0, 65}]] (* Robert G. Wilson v, Jan 31 2011 *)
CoefficientList[Series[x^6/QPochhammer[x^6, x], {x,0,70}], x] (* G. C. Greubel, Nov 03 2019 *)
Join[{0},Table[Count[IntegerPartitions[n][[;;,-1]],6],{n,70}]] (* Harvey P. Dale, Dec 27 2023 *)

my(x='x+O('x^60)); concat([0,0,0,0,0,0], Vec(x^6/prod(m=0,70, 1-x^(m+6)))) \\ G. C. Greubel, Nov 03 2019

def A026799_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^6/product((1-x^(m+6)) for m in (0..70)) ).list()
A026799_list(65) # G. C. Greubel, Nov 03 2019

G.f.: x^6 * Product_{m>=6} 1/(1-x^m).
a(n) = p(n-6) -p(n-7) -p(n-8) +p(n-11) +p(n-12) +p(n-13) -p(n-14) -p(n-15) -p(n-16) +p(n-19) +p(n-20) -p(n-21) for n>0 where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(6*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

A185327 Number of partitions of n into parts >= 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045

Offset: 0

Jason Kimberley, Feb 03 2011

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 7 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. Hence this sequence is essentially the same as A026800.

			The  a(0)=1 empty partition vacuously has each part >= 7.
The  a(7)=1 partition is 7.
The  a(8)=1 partition is 8.
............................
The a(13)=1 partition is 13.
The a(14)=2 partitions are 7+7 and 14.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

2-regular simple graphs with girth at least 7: A185117 (connected), A185227 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), this sequence (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A185327 := func< n | p(n)-p(n-1)-p(n-2)+p(n-5)+2*p(n-7)-p(n-9)-p(n-10)- p(n-11)-p(n-12)+2*p(n-14)+p(n-16)-p(n-19)-p(n-20)+p(n-21) >;

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(1/mul(1-x^(m+7), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

f[1, 1] = f[0, k_] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 7], {n, 0, 65}] (* Robert G. Wilson v, Jan 31 2011 *) (* moved from A026800 by Jason Kimberley, Feb 03 2011 *)
Join[{1},Table[Count[IntegerPartitions[n],?(Min[#]>=7&)],{n,0,70}]] (* _Harvey P. Dale, Oct 16 2011 *)
CoefficientList[Series[1/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+7))) \\ G. C. Greubel, Nov 03 2019

def A185327_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+7)) for m in (0..80)) ).list()
A185327_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=7} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + 2*p(n-7) - p(n-9) - p(n-10) - p(n-11) - p(n-12) + 2*p(n-14) + p(n-16) - p(n-19) - p(n-20) + p(n-21) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010 [moved/copied from A026800 by Jason Kimberley, Feb 03 2011]
This sequence is the Euler transformation of A185117.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(7*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
G.f.: 1 + Sum_{n >= 1} x^(n+6)/Product_{k = 0..n-1} (1 - x^(k+7)). - Peter Bala, Dec 01 2024

cp's OEIS Frontend

A026807 Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n.

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A185325 Number of partitions of n into parts >= 5.

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A124943 Table read by rows: number of partitions of n with k as low median.

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A026797 Number of partitions of n in which the least part is 4.

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A124944 Table, number of partitions of n with k as high median.

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A185326 Number of partitions of n into parts >= 6.

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A026798 Number of partitions of n in which the least part is 5.

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