A225540 Triangle of functions in a size n set for which the sequence of composition powers starts with a length k stem (index) before entering a cycle (period).

1, 1, 4, 21, 6, 148, 84, 24, 1305, 1160, 540, 120, 13806, 17610, 10560, 3960, 720, 170401, 296772, 214410, 104160, 32760, 5040, 2403640, 5536440, 4692576, 2686320, 1115520, 302400, 40320, 38143377, 113680800, 111488328, 72080064, 35637840

Offset: 0

Author

Chad Brewbaker, May 14 2013

Keywords

• nonn
• tabf

Comments

Given a transformation t from the transformation semigroup Tn and a positive integer i, the largest whole number s such that there does not exist an i such that t^s = t^(s+i) under transformation composition with t is the stem length. Terms in t^(s+i) form a cycle under repeated composition with t and do not return to transformations contained in the stem. Permutations by definition have stem length 0.

Examples

			Triangle begins:
       1;
       1;
       4;
      21,      6;
     148,     84,     24;
    1305,   1160,    540,    120;
   13806,  17610,  10560,   3960,   720;
  170401, 296772, 214410, 104160, 32760, 5040;
  ...
		

Links

• Chad Brewbaker, Table of n, a(n) for n = 0..46

Crossrefs

The first column is A006153.

Row sums give A000312.

Cf. A216242.

Programs

Mathematica

nn = 10; Map[Select[#, # > 0 &] &,Table[Range[0, nn]! CoefficientList[      Series[Prepend[Exp[Log[1/(1 - NestList[x Exp[#] &, x Exp[x], nn])]],0][[k + 1]] - Prepend[Exp[Log[1/(1 - NestList[x Exp[#] &, x Exp[x], nn])]], 0][[k]], {x, 0, nn}], x], {k, 1, nn - 1}] // Transpose] // Grid (* Geoffrey Critzer, Feb 13 2022 *)

Ruby

# needs version 1.9+
counting_numbers = Enumerator.new do |yielder|
  (0..1.0/0).each do |number|
    yielder.yield number
  end
end
def trans_mult(transa, transb)
  trans_ret = Array.new
  0.upto(transa.length-1) do |index|
    trans_ret.push(transa[transb[index]])
  end
  return trans_ret
end
def lolipop(trans)
  trans_hash ={}
  trans_hash[trans.clone] =0
  index = 1
  trans_current = trans_mult(trans, trans)
  while  trans_hash[trans_current] == nil
    trans_hash[trans_current.clone] = index
    index = index +1
    trans_current = trans_mult(trans_current, trans)
  end
  cycle_length =trans_hash.size - trans_hash[trans_current]
  return [trans_hash.size, cycle_length]
end
1.upto(10) do |index|
  tran_size =index
  histo_hash = {}
  counting_numbers.take(tran_size).repeated_permutation(tran_size).each { |x|
    size, cycle_length = lolipop(x)
    #tail_length = size-cycle_length
    if (histo_hash[size-cycle_length] == nil)
      histo_hash[size-cycle_length] =1
    else
      histo_hash[size-cycle_length] = histo_hash[size-cycle_length]+1
    end
  }
  puts "#{tran_size}|" + histo_hash.inspect
end