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A298636 Square array T(m,n) = number of ways to draw m-1 horizontal lines [a(i),b(i)] with 0 <= a(i) < b(i) <= n such that if two lines start or end on the same coordinate, no intermediate line crosses this coordinate (see comments); m, n >= 1. 1
1, 1, 1, 1, 3, 1, 1, 6, 9, 1, 1, 10, 36, 23, 1, 1, 15, 100, 181, 53, 1, 1, 21, 225, 845, 775, 115, 1, 1, 28, 441, 2890, 5957, 2956, 241, 1, 1, 36, 784, 8036, 30862, 36148, 10426, 495, 1, 1, 45, 1296, 19278, 122276, 278530, 195934, 34899, 1005, 1, 1, 55, 2025, 41406, 398874, 1560118 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Following the OEIS standard, the array is read by falling antidiagonals, i.e., T(1,1), T(1,2), T(2,1), T(1,3), ....

"Horizontal line [a(i),b(i)]" means a line from (a(i),i) to (b(i),i). "No intermediate line crosses..." means that, if {a(i),b(i)} and {a(j),b(j)} have x in common for some j > i, then for all i < k < j, either a(k) >= x or b(k) <= x.

Equivalently, number of (m-1) X n binary (0,1) matrices where each row has exactly one run of 1's and any two of these runs may not start or end at the same column border, unless no run in the intermediate rows crosses (= extends to both sides of) this border.

This construction is relevant for enumerating the tight pavings defined by Knuth in A285357, see his Christmas Tree Lecture video there.

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

The table starts (cf. "table" link):

  1   1    1     1     1      1     1 ...

  1   3    6    10    15     21    28 ...  (= A000217 = n -> n(n+1)/2)

  1   9   36   100   225    441   784 ...  (= A000537 = A000217^2)

  1  23  181   845  2890   8036 19278...

  1  53  775  5957 30862 122276  ...

  1 115 2956 36148  ...

  ...

Column 2 is A183155.

The T(2,3) = 6 drawings are { [0-1], [0-2], [0-3], [1-2], [1-3], [2-3] }.

The T(3,2) = 9 drawings are { [0-1; 0-1], [0-1; 0-2], [0-1; 1-2], [0-2; 0-1], [0-2, 0-2], [0-2; 1-2], [1-2; 0-1], [1-2; 0-2], [1-2; 1-2] }.

The "no line crosses" condition becomes effective only for m > 3. For m = 4, it excludes drawings like, e.g., [0-1; 0-2; 0-1], [0-1; 0-2; 1-2], ...

Therefore, T(4,2) is less than 3*3*3 = 27: The T(4,2) = 23 drawings are:

{ [0-1; 0-1; 0-1], [0-1; 0-1; 0-2], [0-1; 0-2; 0-2], [0-2; 0-1; 0-1],

  [0-2; 0-1; 0-2], [0-2; 0-2; 0-1], [0-2; 0-2; 0-2], [0-1; 0-1; 1-2],

  [0-2; 0-1; 1-2], [0-2; 0-2; 1-2], [0-1; 1-2; 0-1], [0-1; 1-2; 0-2],

  [0-2; 1-2; 0-1], [0-2; 1-2; 0-2], [0-1; 1-2; 1-2], [0-2; 1-2; 1-2],

  [1-2; 0-1; 0-1], [1-2; 0-1; 0-2], [1-2; 0-2; 0-2], [1-2; 0-1; 1-2],

  [1-2; 1-2; 0-1], [1-2; 1-2; 0-2], [1-2; 1-2; 1-2] }

PROG

(PARI) A298636(m, n, show=0, c=0)={ my(S, N, u=vector(m-1, i, 1)); forvec(a=vector(m-1, i, [0, n-1]), S=Set(a); N=vector(n-1); for(i=1, #a, a[i] && N[a[i]]=if(N[a[i]], concat(N[a[i]], i), i)); forvec(b=vector(m-1, j, [a[j]+1, n]), S=N; for(i=1, #b, b[i]<n && S[b[i]]=if(S[b[i]], concat(S[b[i]], i), i)); for(i=1, #S, #S[i]<2 && next; for(j=2, #T=Set(S[i]), T[j-1]==T[j]-1&&next; for(r=T[j-1]+1, T[j]-1, a[r]>i || b[r]<i || next(4)))); c++; show&&print1(Mat([a~, b~])", "))); c}

CROSSREFS

Cf. A285357.

Cf. A183155, A000537, A000217.

Sequence in context: A158359 A046716 A202605 * A123354 A120247 A235113

Adjacent sequences:  A298633 A298634 A298635 * A298637 A298638 A298639

KEYWORD

nonn,tabl,more

AUTHOR

M. F. Hasler, Jan 23 2018

STATUS

approved

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Last modified November 18 19:59 EST 2019. Contains 329288 sequences. (Running on oeis4.)