#! /usr/bin/env python
#
def stirling1_values ( n_data ):

#*****************************************************************************80
#
## STIRLING1_VALUES returns some values of the Stirling numbers, kind 1.
#
#  Discussion:
#
#    The absolute value of the Stirling number S1(N,M) gives the number
#    of permutations on N objects having exactly M cycles, while the
#    sign of the Stirling number records the sign (odd or even) of
#    the permutations.  For example, there are six permutations on 3 objects:
#
#      A B C   3 cycles (A) (B) (C)
#      A C B   2 cycles (A) (BC)
#      B A C   2 cycles (AB) (C)
#      B C A   1 cycle  (ABC)
#      C A B   1 cycle  (ABC)
#      C B A   2 cycles (AC) (B)
#
#    There are 
#
#      2 permutations with 1 cycle, and S1(3,1) = 2
#      3 permutations with 2 cycles, and S1(3,2) = -3,
#      1 permutation with 3 cycles, and S1(3,3) = 1.
#
#    Since there are N! permutations of N objects, the sum of the absolute 
#    values of the Stirling numbers in a given row, 
#
#      sum ( 1 <= I <= N ) abs ( S1(N,I) ) = N!
#
#  First terms:
#
#    N/M:  1     2      3     4     5    6    7    8
#
#    1     1     0      0     0     0    0    0    0
#    2    -1     1      0     0     0    0    0    0
#    3     2    -3      1     0     0    0    0    0
#    4    -6    11     -6     1     0    0    0    0
#    5    24   -50     35   -10     1    0    0    0
#    6  -120   274   -225    85   -15    1    0    0
#    7   720 -1764   1624  -735   175  -21    1    0
#    8 -5040 13068 -13132  6769 -1960  322  -28    1
#
#    In Mathematica, the function can be evaluated by:
#
#      StirlingS1[n,m]
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    22 February 2015
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Milton Abramowitz and Irene Stegun,
#    Handbook of Mathematical Functions,
#    US Department of Commerce, 1964.
#
#    Stephen Wolfram,
#    The Mathematica Book,
#    Fourth Edition,
#    Wolfram Media / Cambridge University Press, 1999.
#
#  Parameters:
#
#    Input/output, integer N_DATA.  The user sets N_DATA to 0 before the
#    first call.  On each call, the routine increments N_DATA by 1, and
#    returns the corresponding data; when there is no more data, the
#    output value of N_DATA will be 0 again.
#
#    Output, integer N, M, the arguments of the function.
#
#    Output, integer F, the value of the function.
#
  import numpy as np

  n_max = 16

  f_vec = np.array ( ( \
           0, \
           1, \
          -3, \
          11, \
         -50, \
         274, \
       -1764, \
       13068, \
     -109584, \
     1026576, \
      -13132, \
        6769, \
       -1960, \
         322, \
         -28, \
           1 ))

  m_vec = np.array ( ( \
     2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 5, 6, 7, 8 ))

  n_vec = np.array ( ( \
     1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 8, 8, 8, 8, 8, 8 ))

  if ( n_data < 0 ):
    n_data = 0

  if ( n_max <= n_data ):
    n_data = 0
    n = 0
    m = 0
    f = 0
  else:
    n = n_vec[n_data]
    m = m_vec[n_data]
    f = f_vec[n_data]
    n_data = n_data + 1

  return n_data, n, m, f

def stirling1_values_test ( ):

#*****************************************************************************80
#
## STIRLING1_VALUES_TEST demonstrates the use of STIRLING1_VALUES.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    22 February 2015
#
#  Author:
#
#    John Burkardt
#
  import platform

  print ( '' )
  print ( 'STIRLING1_VALUES_TEST:' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  STIRLING1_VALUES stores values of the STIRLING1 function.' )
  print ( '' )
  print ( '             N             M        STIRLING1(N,M)' )
  print ( '' )

  n_data = 0

  while ( True ):

    n_data, n, m, f = stirling1_values ( n_data )

    if ( n_data == 0 ):
      break

    print ( '  %12d  %12d  %12d' % ( n, m, f ) )
#
#  Terminate.
#
  print ( '' )
  print ( 'STIRLING1_VALUES_TEST:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( )
  stirling1_values_test ( )
  timestamp ( )