A332773 -id:A332773 - cp's OEIS frontend

A332752 The number of permutations of {1,1,1,1,2,2,2,2,...,n,n,n,n} such that each quadruple of k's (k=1..n) is equally spaced with b(k) other elements in between, and b(1) >= b(2) >= ... >= b(n).

1, 1, 4, 16, 110, 544, 5444, 32520, 385776, 3282108, 40916528, 354328560, 7200045216, 67347823160, 1182323197504, 18086875471594, 358787259407482, 4564034487662420

Offset: 0

Seiichi Manyama, Feb 22 2020

			In case of n = 1.
     |              | b(1)
-----+--------------+------
   1 | [1, 1, 1, 1] | [0] *
In case of n = 2.
     |                          | b(1),b(2)
-----+--------------------------+----------
   1 | [2, 2, 2, 2, 1, 1, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1, 2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 1, 2, 1, 2, 1, 2] | [1, 1]
   4 | [1, 1, 1, 1, 2, 2, 2, 2] | [0, 0]
In case of n = 3.
     |                                      | b(1),b(2),b(3)
-----+--------------------------------------+---------------
   1 | [3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1] | [0, 0, 0]
   2 | [3, 3, 3, 3, 2, 1, 2, 1, 2, 1, 2, 1] | [1, 1, 0]
   3 | [3, 3, 3, 3, 1, 2, 1, 2, 1, 2, 1, 2] | [1, 1, 0]
   4 | [3, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2] | [0, 0, 0]
   5 | [3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1] | [2, 2, 2]
   6 | [3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2] | [2, 2, 2]
   7 | [2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1] | [2, 2, 2]
   8 | [1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2] | [2, 2, 2]
   9 | [2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3] | [2, 2, 2]
  10 | [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3] | [2, 2, 2]
  11 | [2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1] | [0, 0, 0]
  12 | [1, 1, 1, 1, 3, 3, 3, 3, 2, 2, 2, 2] | [0, 0, 0]
  13 | [2, 2, 2, 2, 1, 1, 1, 1, 3, 3, 3, 3] | [0, 0, 0]
  14 | [2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 3, 3] | [1, 1, 0]
  15 | [1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 3] | [1, 1, 0]
  16 | [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3] | [0, 0, 0]
* (strongly decreasing)

Column 4 of A332762.

Cf. A104430, A261517 (strongly decreasing), A285698, A322178, A332748, A332773, A332783, A332784.

a(10)-a(17) from Max Alekseyev, Sep 27 2023

A332762 Square array T(n,k), n >= 0, k >= 2, read by antidiagonals, where T(n,k) is the number of permutations of {k 1's, k 2's, ..., k n's} with the property that j's are equally spaced for j=1..n and the interval of j+1 is less than or equal to the interval of j for j=1..n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 4, 33, 1, 1, 4, 18, 329, 1, 1, 4, 16, 124, 3825, 1, 1, 4, 16, 110, 738, 57293, 1, 1, 4, 16, 104, 544, 7464, 977581, 1, 1, 4, 16, 104, 508, 5444, 55890, 19619645, 1, 1, 4, 16, 104, 484, 5136, 32520, 668778, 442155529

Offset: 0

Seiichi Manyama, Feb 22 2020

			Square array begins:
      1,    1,    1,    1,    1, ...
      1,    1,    1,    1,    1, ...
      5,    4,    4,    4,    4, ...
     33,   18,   16,   16,   16, ...
    329,  124,  110,  104,  104, ...
   3825,  738,  544,  508,  484, ...
  57293, 7464, 5444, 5136, 4968, ...

Columns k=2..5 give A322178, A332748, A332752, A332773.
T(n,n) gives A332784.
T(n,n+1) gives A332783.

A332783 The number of permutations of {(n+1) 1's, (n+1) 2's, ..., (n+1) n's} with the property that k's are equally spaced for k=1..n and the interval of k+1 is less than or equal to the interval of k for k=1..n-1.

Original entry on oeis.org

1, 4, 16, 104, 484, 4848, 25104, 300336, 2335296, 27953952, 198725952, 4731323904, 33020828928, 606237831936, 8936541384192, 174694058933760, 1628654065588224, 56338295740213248, 545177455792662528, 20766878061520306176, 340162958990367645696

Offset: 1

Seiichi Manyama, Feb 23 2020

nonn

			Define the interval of k as b(k).
In case of n = 1.
     |        | b(1)
-----+--------+-----
   1 | [1, 1] | [0]
In case of n = 2.
     |                    | b(1),b(2)
-----+--------------------+----------
   1 | [2, 2, 2, 1, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 1, 2, 1, 2] | [1, 1]
   4 | [1, 1, 1, 2, 2, 2] | [0, 0]
In case of n = 3.
     |                                      | b(1),b(2),b(3)
-----+--------------------------------------+---------------
   1 | [3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1] | [0, 0, 0]
   2 | [3, 3, 3, 3, 2, 1, 2, 1, 2, 1, 2, 1] | [1, 1, 0]
   3 | [3, 3, 3, 3, 1, 2, 1, 2, 1, 2, 1, 2] | [1, 1, 0]
   4 | [3, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2] | [0, 0, 0]
   5 | [3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1] | [2, 2, 2]
   6 | [3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2] | [2, 2, 2]
   7 | [2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1] | [2, 2, 2]
   8 | [1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2] | [2, 2, 2]
   9 | [2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3] | [2, 2, 2]
  10 | [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3] | [2, 2, 2]
  11 | [2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1] | [0, 0, 0]
  12 | [1, 1, 1, 1, 3, 3, 3, 3, 2, 2, 2, 2] | [0, 0, 0]
  13 | [2, 2, 2, 2, 1, 1, 1, 1, 3, 3, 3, 3] | [0, 0, 0]
  14 | [2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 3, 3] | [1, 1, 0]
  15 | [1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 3] | [1, 1, 0]
  16 | [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3] | [0, 0, 0]

Cf. A104442, A332762, A332784, A322178, A332748, A332752, A332773.

def search(a, num, d, k, n)
  if num == 0
    @cnt += 1
  else
    (k * n - k + 1).times{|i|
      if a[i] == 0
        (i + d + 1..k * n - k + 1).each{|j|
          if (k - 1) * j - (k - 2) * i < k * n
            if (1..k - 1).all?{|m| a[m * j - (m - 1) * i] == 0}
              (0..k - 1).each{|m| a[m * j - (m - 1) * i] = num}
              search(a, num - 1, j - i - 1, k, n)
              (0..k - 1).each{|m| a[m * j - (m - 1) * i] = 0}
            end
          end
        }
      end
    }
  end
end
def A(k, n)
  a = [0] * k * n
  @cnt = 0
  search(a, n, 0, k, n)
  @cnt
end
def A332783(n)
  (1..n).map{|i| A(i + 1, i)}
end
p A332783(5)

a(9)-a(17) from Bert Dobbelaere, Mar 08 2020

a(18)-a(21) from Max Alekseyev, Sep 26 2023

A332784 The number of permutations of {n 1's, n 2's,...,n n's} with the property that b(1) >= b(2) >= ... >= b(n), where n k's are skipped by b(k) for k=1..n.

Original entry on oeis.org

5, 18, 110, 508, 4968, 25824, 305376, 2375616, 28316832, 202354752, 4771240704, 33499830528, 612464852736, 9023719675392, 176001733301760, 1649576855476224, 56693983168309248, 551579829498390528, 20888523161929138176, 342595860998544285696

Offset: 2

Seiichi Manyama, Feb 23 2020

nonn

			In case of n = 2.
     |              | b(1),b(2)
-----+--------------+----------
   1 | [2, 2, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 2, 1] | [2, 0]
   4 | [1, 2, 1, 2] | [1, 1]
   5 | [1, 1, 2, 2] | [0, 0]
In case of n = 3.
     |                             | b(1),b(2),b(3)
-----+-----------------------------+---------------
   1 | [3, 3, 3, 2, 2, 2, 1, 1, 1] | [0, 0, 0]
   2 | [3, 3, 3, 2, 1, 2, 1, 2, 1] | [1, 1, 0]
   3 | [3, 3, 3, 1, 2, 1, 2, 1, 2] | [1, 1, 0]
   4 | [3, 3, 3, 1, 1, 1, 2, 2, 2] | [0, 0, 0]
   5 | [3, 2, 1, 3, 2, 1, 3, 2, 1] | [2, 2, 2]
   6 | [3, 1, 2, 3, 1, 2, 3, 1, 2] | [2, 2, 2]
   7 | [1, 3, 3, 3, 1, 2, 2, 2, 1] | [3, 0, 0]
   8 | [2, 3, 1, 2, 3, 1, 2, 3, 1] | [2, 2, 2]
   9 | [1, 3, 2, 1, 3, 2, 1, 3, 2] | [2, 2, 2]
  10 | [2, 1, 3, 2, 1, 3, 2, 1, 3] | [2, 2, 2]
  11 | [1, 2, 3, 1, 2, 3, 1, 2, 3] | [2, 2, 2]
  12 | [2, 2, 2, 3, 3, 3, 1, 1, 1] | [0, 0, 0]
  13 | [1, 1, 1, 3, 3, 3, 2, 2, 2] | [0, 0, 0]
  14 | [1, 2, 2, 2, 1, 3, 3, 3, 1] | [3, 0, 0]
  15 | [2, 2, 2, 1, 1, 1, 3, 3, 3] | [0, 0, 0]
  16 | [2, 1, 2, 1, 2, 1, 3, 3, 3] | [1, 1, 0]
  17 | [1, 2, 1, 2, 1, 2, 3, 3, 3] | [1, 1, 0]
  18 | [1, 1, 1, 2, 2, 2, 3, 3, 3] | [0, 0, 0]

Cf. A104442, A332762, A332783, A322178, A332748, A332752, A332773.

Conjecture: a(n) = A332783(n) + (n-1)!.

a(9)-a(17) from Bert Dobbelaere, Mar 08 2020

a(18)-a(21) from Max Alekseyev, Sep 26 2023

cp's OEIS Frontend

A332752 The number of permutations of {1,1,1,1,2,2,2,2,...,n,n,n,n} such that each quadruple of k's (k=1..n) is equally spaced with b(k) other elements in between, and b(1) >= b(2) >= ... >= b(n).

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A332762 Square array T(n,k), n >= 0, k >= 2, read by antidiagonals, where T(n,k) is the number of permutations of {k 1's, k 2's, ..., k n's} with the property that j's are equally spaced for j=1..n and the interval of j+1 is less than or equal to the interval of j for j=1..n-1.

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A332783 The number of permutations of {(n+1) 1's, (n+1) 2's, ..., (n+1) n's} with the property that k's are equally spaced for k=1..n and the interval of k+1 is less than or equal to the interval of k for k=1..n-1.

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A332784 The number of permutations of {n 1's, n 2's,...,n n's} with the property that b(1) >= b(2) >= ... >= b(n), where n k's are skipped by b(k) for k=1..n.

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