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A141058
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Pats by first entry.
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0
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1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 2, 3, 4, 5, 0, 5, 6, 7, 10, 14, 0, 14, 15, 15, 18, 28, 42, 0, 42, 42, 38, 40, 51, 84, 132, 0, 132, 126, 107, 103, 115, 154, 264, 429, 0, 429, 396, 322, 292, 299, 350, 486, 858, 1430, 0, 1430, 1287, 1014, 882, 852, 915, 1110, 1584, 2860, 4862
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OFFSET
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0,9
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COMMENTS
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T(n,k) is the number of pats on [0,n] with first entry k. Pats are defined recursively in the Oakley/Wisner reference. Briefly, a one-entry permutation is a pat and a two-or-more-entry permutation p on any set of integers is a pat iff (i) there is a unique way to split p as the concatenation of nonempty permutations p_1 and p_2 such that all entries in p_1 exceed all entries in p_2, and (ii) reverse(p1) and reverse(p2) are pats. Thus 21 and 43 are pats but 12 is not and p = 3412 is a pat using p1 = 34 and p2 = 12. Pats on [1,n+1] (considered by Oakley/Wisner in the definition of flexagons) correspond to pats on [0,n] by subtracting 1 from each entry.
Also, pats on [0,n] with last entry k correspond to pats with first entry n-k under the reverse-complement operation on permutations.
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LINKS
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C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly 64 (1957), 143-154.
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FORMULA
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G.f.: Sum_{n>=0,k>=0} T(n,k)*x^n*y^k = (1 + x * y * CatalanGF(x * y))/(1 - x^2 * y * CatalanGF(x) * CatalanGF(x * y)) where CatalanGF(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. for the Catalan numbers A000108.
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EXAMPLE
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T(4,2)=3 counts 24301, 23140, 21430.
Table begins
1
0...1
0...1...1
0...1...2...2
0...2...3...4...5
0...5...6...7..10..14
0..14..15..15..18..28..42
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MATHEMATICA
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a[0, 0]=1; a[n_, k_]/; n>=1 && 0<=k<=n := a[n, k] = (* count by splitting point in condition (i) *) Sum[a[i, n-k]CatalanNumber[n-i-1], {i, n-k, n-1}]; Table[a[n, k], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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The Catalan numbers A000108 appear as row sums and in the second column and on the main diagonal.
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KEYWORD
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AUTHOR
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STATUS
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approved
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