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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 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.)