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A309053 Triangular array T read by rows: T(r,c) is the number of double permutations of the integers from 1 to r which have exactly c different values visible when viewed from the left, in the sense that a higher number hides a lower one. 0
1, 0, 1, 0, 1, 3, 0, 4, 17, 15, 0, 36, 181, 254, 105, 0, 576, 3220, 5693, 3966, 945, 0, 14400, 86836, 177745, 161773, 67251, 10395, 0, 518400, 3313296, 7527688, 8134513, 4524085, 1248483, 135135 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Consider r rectangular cards stacked in a pile with their left and lower edges aligned. Each is of a different color and their widths and heights are independent permutations of the integers 1, 2, ..., r. Then the sequence gives the number of ways in which exactly c colors may be seen, where 0 <= c <= r. The values are entries in a triangular table read from left to right along successive rows from the top, each row giving the value of r and each column giving the value of c. Including a row in the triangle for r = 0 and treating the values as a list a(n) starting with n = 1,  n = r(r+1)/2 + c + 1.

For example, r = 2. If the widths of the cards from the top of the stack are 1,2 and the heights are 1,2 then two colors are seen; if the widths are 1,2 and the heights are 2,1 then two colors are seen; if 2,1 and 1,2 then two colors are seen; if 2,1 and 2,1 then only one color is seen. Thus the values for c = 1 and c = 2 are 1 and 3 respectively, i.e., a(5) = 1 and a(6) = 3.

The triangle up to r = 7 is:

  r\c   0      1       2       3       4       5       6      7

  0     1

  1     0      1

  2     0      1       3

  3     0      4      17      15

  4     0     36     181     254     105

  5     0    576    3220    5693    3966     945

  6     0  14400   86836  177745  161773   67251   10395

  7     0 518400 3313296 7527688 8134513 4524085 1248483 135135

The sum of row r in the table is (r!)^2 and T(r,1) for r > 0 is ((r-1)!)^2.

LINKS

Table of n, a(n) for n=0..35.

PROG

(Basic)

r=5

fr=1

for i=2 to r : fr=fr*i : next i            ' fr=r!

dim perm(fr, r), a(fr, r), b(r), count(r), p(r)

for i=1 to fr : for j=1 to r : a(i, j)=0 : next j : next i

for i=1 to r : count(i)=0 : next i

'*** now derive successive permutations p() and populate rows of perm()

for k=0 to fr-1

   for i=1 to r : p(i)=i : next i

   f=1

   for j=2 to r

      f=f*(j-1)

      a=int(k/f)

      i=a mod j

      x=p(j-i) : p(j-i)=p(j) : p(j)=x

   next j

   for i=1 to r

      perm(k+1, i)=p(i)

   next i

next k

'***

'*** now determine which numbers are visible for each permutation and

'     put in a()

for k=1 to fr

   max=perm(k, 1)

   a(k, perm(k, 1))=1

   for i=2 to r

      if perm(k, i)>max then max=perm(k, i) : a(k, perm(k, i))=1

   next i

next k

'***

'*** now determine which numbers [b()], and how many [count()], are

'     visible for each combination of permutations

for i=1 to fr

   for j=1 to fr

      tb=0

      for k=1 to r

         b(k)=0 : if a(i, k)=1 or a(j, k)=1 then b(k)=1

         tb=tb+b(k)

      next k

      count(tb)=count(tb)+1

   next j

next i

'***

for c=1 to r

   print c; "   "; count(c)

next c

CROSSREFS

Row sums and T(r,1) for r > 0 give A001044.

Main diagonal gives A001147.

Cf. A132393, giving the analogous table for a single permutation, i.e., cards varying only by width or by height.

Sequence in context: A074171 A180657 A094665 * A052439 A261239 A261214

Adjacent sequences:  A309050 A309051 A309052 * A309054 A309055 A309056

KEYWORD

nonn,tabl,more

AUTHOR

Ian Duff, Jul 09 2019

STATUS

approved

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Last modified April 3 17:03 EDT 2020. Contains 333197 sequences. (Running on oeis4.)