A239497 -id:A239497 - cp's OEIS frontend

A239498 Number of partitions p of n such that if h = 2*min(p), then h is an (h,1)-separator of p; see Comments.

0, 0, 1, 0, 0, 2, 0, 1, 3, 1, 2, 5, 4, 4, 8, 7, 9, 15, 15, 18, 23, 26, 32, 43, 47, 57, 72, 80, 98, 120, 138, 163, 198, 227, 267, 323, 372, 438, 517, 596, 696, 818, 944, 1098, 1282, 1477, 1711, 1989, 2285, 2637, 3049, 3496, 4023, 4633, 5303, 6080, 6976, 7968

Offset: 1

Clark Kimberling, Mar 24 2014

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

			a(9) counts these partitions: 63, 4212, 212121.

Cf. A239497, A118096, A239500, A239501, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 29, 32, 36, 39, 43, 48, 53, 58, 65, 70, 78, 85, 93, 101, 112, 120, 132, 143, 156, 168, 184, 198, 216, 233, 253, 273, 298, 320, 348, 376, 407, 439

Offset: 1

Clark Kimberling, Mar 24 2014

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

			a(12) counts these partitions: 84, 4431, 4422.

Cf. A239497, A239498, A118096, A239501, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A239501 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,1)-separator of p; see Comments.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 3, 3, 2, 2, 3, 5, 4, 8, 4, 5, 9, 6, 13, 10, 11, 15, 14, 17, 16, 20, 21, 26, 29, 30, 33, 36, 35, 41, 47, 47, 61, 61, 66, 71, 73, 85, 88, 98, 102, 114, 122, 131, 148, 154, 163, 182, 188, 205, 220, 231, 249, 271, 293, 306, 338, 359

Offset: 1

Clark Kimberling, Mar 24 2014

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

			a(11) counts these partitions: 4313, 4232, 321212.

Cf. A239497, A239498, A239499, A239500, A239482.

z = 35; t1 = Table[Count[IntegerPartitions[n],  p_ /; 2 Count[p, Min[p]] == Length[p]], {n, 1, z}]  (* A239497 *)
t2 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2 Min[p]] == Length[p]], {n, 1, z}] (* A239498 *)
t3 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p]] == Length[p]], {n, 1, z}] (* A118096 *)
t4 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Length[p]] == Length[p]], {n, 1, z}] (* A239500 *)
t5 = Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, Max[p] - Min[p]] == Length[p]], {n, 1, z}]  (* A239501 *)

A386360 Number of partitions of n such that the least part occurs exactly (1/3)*(number of parts) times.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 27, 33, 41, 50, 63, 75, 93, 111, 136, 163, 198, 235, 285, 337, 406, 479, 574, 676, 806, 948, 1124, 1318, 1557, 1822, 2147, 2505, 2940, 3424, 4006, 4657, 5431, 6299, 7329, 8483, 9843, 11372, 13163, 15177, 17527, 20175

Offset: 1

Seiichi Manyama, Jul 19 2025

nonn

Cf. A000005, A239497, A386361.

Cf. A237825.

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(n, k)
  cnt = 0
  partition(n, 1, n).each{|ary|
    cnt += 1 if k * ary.count(ary.min) == ary.size
  }
  cnt
end
def A386360(n)
  (1..n).map{|i| A(i, 3)}
end
p A386360(40)

A386361 Number of partitions of n such that the least part occurs exactly (1/4)*(number of parts) times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 12, 16, 20, 26, 32, 41, 50, 62, 76, 93, 112, 136, 164, 197, 237, 283, 339, 403, 480, 569, 676, 799, 945, 1113, 1314, 1543, 1815, 2125, 2494, 2912, 3407, 3970, 4632, 5384, 6266, 7268, 8438, 9766, 11310, 13063, 15097, 17402

Offset: 1

Seiichi Manyama, Jul 19 2025

nonn

Cf. A000005, A239497, A386360.

Cf. A237826.

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(n, k)
  cnt = 0
  partition(n, 1, n).each{|ary|
    cnt += 1 if k * ary.count(ary.min) == ary.size
  }
  cnt
end
def A386361(n)
  (1..n).map{|i| A(i, 4)}
end
p A386361(40)

cp's OEIS Frontend

A239498 Number of partitions p of n such that if h = 2*min(p), then h is an (h,1)-separator of p; see Comments.

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Mathematica

A239500 Number of partitions p of n such that if h = (number of parts of p), then h is an (h,1)-separator of p; see Comments.

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Mathematica

A239501 Number of partitions p of n such that if h = max(p) - min(p), then h is an (h,1)-separator of p; see Comments.

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Mathematica

A386360 Number of partitions of n such that the least part occurs exactly (1/3)*(number of parts) times.

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Ruby

A386361 Number of partitions of n such that the least part occurs exactly (1/4)*(number of parts) times.

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Ruby