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A362563
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Triangle T(n, k) read by rows, where T(n, k) is the number of {123,132}-avoiding parking functions of size n with k active sites, for 2 <= k <= n+1.
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1
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1, 1, 2, 1, 3, 4, 3, 5, 8, 8, 8, 14, 17, 20, 16, 24, 40, 49, 50, 48, 32, 75, 123, 147, 151, 136, 112, 64, 243, 393, 465, 473, 432, 352, 256, 128, 808, 1294, 1519, 1540, 1409, 1176, 880, 576, 256, 2742, 4358, 5087, 5144, 4721, 3986, 3088, 2144, 1280, 512
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OFFSET
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1,3
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COMMENTS
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Consider a parking function of size n that avoids both 123 and 132.
Such a parking function can be represented as a labeled Dyck path (using steps N = (0, 1) and E = (1, 0) staying weakly above y = x), where the north steps are labeled with 1, 2, ..., n, and where consecutive north steps have increasing labels.
An active site is a point where the parking function's corresponding Dyck path touches y = x.
T(n, k) is the number of parking functions of size n with exactly k active sites.
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LINKS
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FORMULA
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T(n, k) = 0 if k < 2 or k > n+1
T(1, 2) = T(2, 2) = 1.
T(2, 3) = 2.
For n > 2, T(n, k) = 2*T(n-1, k-1) + Sum_{j=k-1..n-1} T(n-2, j).
Sum_{k=2..n+1} T(n, k) = T(n+2, 2) = A000958(n+1).
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EXAMPLE
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Triangle T(n, k) begins:
1;
1, 2;
1, 3, 4;
3, 5, 8, 8;
8, 14, 17, 20, 16;
24, 40, 49, 50, 48, 32;
75, 123, 147, 151, 136, 112, 64;
243, 393, 465, 473, 432, 352, 256, 128;
808, 1294, 1519, 1540, 1409, 1176, 880, 576, 256;
2742, 4358, 5087, 5144, 4721, 3986, 3088, 2144, 1280, 512;
...
The eight {123,132}-avoiding parking functions of size 3 are 211, 212, 213, 221, 231, 311, 312, and 321.
In block notation:
211 is {2,3},{1},{} -> NNENEE, which has 2 active sites;
212 is {2},{1, 3},{} -> NENNEE, which has 3 active sites;
213 is {2},{1},{3} -> NENENE, which has 4 active sites;
221 is {3},{1,2},{} -> NENNEE, which has 3 active sites;
231 is {3},{1},{2} -> NENENE, which has 4 active sites;
311 is {2,3},{},{1} -> NNEENE, which has 3 active sites;
312 is {2},{3},{1} -> NENENE, which has 4 active sites;
321 is {3},{2},{1} -> NENENE, which has 4 active sites.
So T(3,2) = 1, T(3,3) = 3, T(3,4) = 4.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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