A231428 -id:A231428 - cp's OEIS frontend

A249615 Number of non-singleton blocks in the n-th set partition (A231428).

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2

Offset: 0

Tilman Piesk, Nov 02 2014

nonn

			The 6th set partition has 2 non-singleton blocks {1,4} and {2,3}. So a(6) = 2.

Tilman Piesk, Table of n, a(n) for n = 0..9999
Tilman Piesk, Illustrated list of the first 52 equivalence relations
Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)

Cf. A231428, A249616

A249616 Number of elements in non-singleton blocks in the n-th set partition (A231428).

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 4, 2, 4, 3, 2, 4, 3, 3, 4, 2, 4, 4, 4, 5, 2, 4, 4, 4, 5, 3, 5, 2, 4, 4, 4, 5, 3, 5, 3, 5, 4, 2, 4, 4, 4, 5, 3, 5, 3, 5, 4, 3, 5, 4, 4, 5, 2, 4, 4, 4, 5, 4, 6, 4, 6, 5, 4, 6, 5, 5, 6, 2, 4, 4, 4, 5, 4, 6, 4, 6, 5, 4, 6, 5, 5, 6

Offset: 0

Tilman Piesk, Nov 02 2014

nonn

			The 6th set partition has the non-singleton blocks {1,4} and {2,3}, and they contain 4 elements. So a(6) = 4.

Tilman Piesk, Table of n, a(n) for n = 0..9999
Tilman Piesk, Illustrated list of the first 52 equivalence relations
Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)

Cf. A231428, A249615.

A249617 Integer partition (A194602) of the n-th set partition (A231428).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 1, 3, 2, 2, 4, 1, 3, 3, 3, 5, 1, 3, 3, 3, 5, 2, 5, 1, 3, 3, 3, 5, 2, 5, 2, 5, 4, 1, 3, 3, 3, 5, 2, 5, 2, 5, 4, 2, 5, 4, 4, 6, 1, 3, 3, 3, 5, 3, 7, 3, 7, 5, 3, 7, 5, 5, 8, 1, 3, 3, 3, 5, 3, 7, 3, 7, 5, 3, 7, 5, 5, 8

Offset: 0

Tilman Piesk, Nov 02 2014

nonn

			The 6th set partition has 2 non-singleton blocks {1,4} and {2,3}, each with 2 elements. This corresponds to the integer partition 2+2, which is the 3rd in the infinite order defined by A194602. So a(6) = 3.

Tilman Piesk, Table of n, a(n) for n = 0..9999
Tilman Piesk, Illustrated list of the first 52 equivalence relations
Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)

A249618 Set partition (A231428) corresponding to the n-th finite permutation (A055089).

Original entry on oeis.org

0, 1, 3, 4, 4, 2, 10, 11, 13, 14, 14, 12, 13, 14, 7, 9, 8, 14, 14, 12, 9, 5, 14, 6, 37, 38, 40, 41, 41, 39, 47, 48, 50

Offset: 0

Tilman Piesk, Nov 02 2014

			The 23rd permutation is (1 4)(2 3) in cycle notation, and the corresponding set partition is {{1,4},{2,3}}, which is the 6th in the infinite order defined by A231428. So a(23) = 6.

Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)

A261195 Encoded symmetrical square binary matrices.

Original entry on oeis.org

0, 1, 6, 7, 16, 17, 22, 23, 40, 41, 46, 47, 56, 57, 62, 63, 384, 385, 390, 391, 400, 401, 406, 407, 424, 425, 430, 431, 440, 441, 446, 447, 576, 577, 582, 583, 592, 593, 598, 599, 616, 617, 622, 623, 632, 633, 638, 639, 960, 961, 966, 967, 976, 977, 982, 983

Offset: 0

Philippe Beaudoin, Aug 11 2015

nonn

We encode an n X n binary matrix reading it antidiagonal by antidiagonal, starting from the least significant bit. A given entry in the sequence therefore represents the infinite family of n X n matrices that can be obtained by adding zero antidiagonals. All of these matrices are symmetrical. This encoding makes it possible to obtain a sequence rather than a table.

			391 = 0b110000111 encodes all square matrices with the first four antidiagonals equal to ((1), (1, 1), (0, 0, 0), (0, 1, 1, 0)), for example, the 3 X 3 matrix:
  1 1 0
  1 0 1
  0 1 0
and the 4 X 4 matrix:
  1 1 0 0
  1 0 1 0
  0 1 0 0
  0 0 0 0
and all larger square matrices constructed in the same way. Since 391 is in the sequence, all these matrices are symmetrical.

Philippe Beaudoin, Table of n, a(n) for n = 0..10000

Cf. A231428, A261194.

b[n_] := Select[ Tuples[{0, 1}, n], # == Reverse@ # &]; FromDigits[#, 2]& /@ Join @@@ Tuples[ b/@ Range[7, 1, -1]] (* Giovanni Resta, Aug 12 2015 *)

a((2n+1)*2^(k-1)) = a(n*2^k) + a(2^(k-1)) for n >= 0 and k >= 1. - Eric Werley, Sep 13 2015

A261194 Encoded square binary matrices representing an idempotent relation.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 16, 17, 18, 19, 20, 21, 23, 25, 27, 33, 37, 49, 53, 65, 67, 73, 75, 81, 83, 89, 91, 141, 144, 145, 148, 149, 153, 154, 155, 157, 159, 181, 209, 217, 219, 272, 273, 274, 275, 283, 291, 305, 307, 308, 309, 311, 337, 339, 347, 513, 517, 529

Offset: 0

Philippe Beaudoin, Aug 11 2015

nonn

We encode an n X n binary matrix reading it antidiagonal by antidiagonal, starting from the least significant bit. A given entry in the sequence therefore represents the infinite family of n X n matrices that can be obtained by adding zero antidiagonals. All of these matrices represent idempotent relations. This encoding makes it possible to obtain a sequence rather than a table.

			For example, 148 = 0b10010100 encodes all square matrices with the first four antidiagonals equal to ((0), (0, 1), (0, 1, 0), (0, 1, 0, 0)). For example the 3 X 3 matrix:
  0 1 0
  0 1 0
  0 1 0
and the 4 X 4 matrix:
  0 1 0 0
  0 1 0 0
  0 1 0 0
  0 0 0 0
and all larger square matrices constructed in the same way. Since 148 is in the sequence, all these matrices are idempotent.

Philippe Beaudoin, Table of n, a(n) for n = 0..6672

Cf. A121337, A231428, A261195.

def getBitIndex(i, j):
  return (i+j)*(i+j+1)//2 + j
def getBit(mat, i, j):
  return (mat >> getBitIndex(i, j)) & 1
def setBit(mat, i, j):
  return mat | (1 << getBitIndex(i, j))
def noBitLeft(mat, i, j):
  return mat >> getBitIndex(i, j) == 0
def squarematrix(mat):
  result = 0;
  i = 0
  while True:
    if noBitLeft(mat, i, 0):
      return result
    j = 0;
    while True:
      if noBitLeft(mat, 0, j):
        break
      k = 0
      while True:
        if noBitLeft(mat, i, k):
          break
        if getBit(mat, i, k) & getBit(mat, k, j):
          result = setBit(result, i, j)
          break
        k += 1
      j += 1
    i += 1
  return result
index = 0
mat = 0
while True:
  if mat == squarematrix(mat):
    print(index, mat)
    index += 1
  mat += 1

cp's OEIS Frontend

A249615 Number of non-singleton blocks in the n-th set partition (A231428).

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A249616 Number of elements in non-singleton blocks in the n-th set partition (A231428).

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A249617 Integer partition (A194602) of the n-th set partition (A231428).

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A249618 Set partition (A231428) corresponding to the n-th finite permutation (A055089).

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A261195 Encoded symmetrical square binary matrices.

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Mathematica

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A261194 Encoded square binary matrices representing an idempotent relation.

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Python