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A098906
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Triangle read by rows: T(n,k) is the number of down-up permutations on [n] with k left-to-right maxima.
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0
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1, 1, 1, 1, 2, 3, 5, 8, 3, 16, 30, 15, 61, 121, 75, 15, 272, 588, 420, 105, 1385, 3128, 2478, 840, 105, 7936, 18960, 16380, 6300, 945, 50521, 124921, 115350, 51030, 11025, 945, 353792, 911328, 893640, 429660, 103950, 10395, 2702765, 7158128, 7365633
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| T(n,k)=0 unless 1 <= k <= (n+1)/2.
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LINKS
| L. Carlitz and R. Scoville, Enumeration of up-down permutations by upper records, Monatshefte für Mathematik, 79 (1975) 3-12.
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FORMULA
| The even-indexed rows have g.f. A(x, y):=Sum_{1<=k<=n}a(n, k)x^(2n)*y^k satisfying the functional equation A(x, y)(1+x*y^2) = x*y(1+(y+1)A(x, y+2)). The odd-indexed rows have g.f. B(x, y):=Sum_{1<=k<=n}b(n, k)x^(2n-1)*y^k satisfying the slightly different equation B(x, y)(1+x*(y+1)^2) = x*y(1+(y+1)B(x, y+2)). The recurrence relations underlying these functional equations are given in the Mathematica code below.
GF (see Carlitz and Scoville link): 1+Sum_{n>=1,1<=k<=n}T(2n,k)x^(2n)/(2n)!*y^k = (sec x)^y,
Sum_{n>=1,1<=k<=n}T(2n-1,k)x^(2n-2)/(2n-2)!y^k = y(sec x)^(1+y). - David Callan Nov 21 2011
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EXAMPLE
| Table begins
\ k..1....2....3......
n
1 |..1
2 |..1
3 |..1 1
4 |..2....3
5 |..5....8....3
6 |.16...30...15
7 |.61..121...75...15
8 |272..588..420..105
For example, w = 21534 has 2 left-to-right maxima: w_1 = 2 and w_3 = 5.
T(4,2) = 3 because 2143, 3142, 3241 each have 2 left-to-right maxima.
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MATHEMATICA
| Clear[a, b] EvenMultiplier[k_, j_]/; j<=k-2 := 0; EvenMultiplier[k_, j_]/; j>=k-1 := (2^(j+1-k) (Binomial[j, k-2]+Binomial[j+1, k-1])); a[1, 1]=1; a[n_, 0]:=0; a[n_, k_]/; 1<=k<=n && n>1 := a[n, k] = Sum[EvenMultiplier[k, j]a[n-1, j], {j, k-1, n-1}]; OddMultiplier[k_, j_]:=EvenMultiplier[k, j]-If[j==k-1, 2, 0]-If[j==k, 1, 0]; b[1, 1]=1; b[n_, 0]:=0; b[n_, k_]/; 1<=k<=n && n>1 := b[n, k] = Sum[OddMultiplier[k, j]b[n-1, j], {j, k-1, n-1}] Flatten[Table[{ Table[b[n, k], {k, n}], Table[a[n, k], {k, n}] }, {n, 7} ], 1]
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CROSSREFS
| Row sums are the up-down numbers (A000111), as is column k=1. Topmost entries in each column form the double factorials (A001147). The even indexed rows form A085734.
Sequence in context: A003893 A152303 A064737 * A007887 A105472 A030132
Adjacent sequences: A098903 A098904 A098905 * A098907 A098908 A098909
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KEYWORD
| nonn,tabl
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AUTHOR
| David Callan (callan(AT)stat.wisc.edu), Nov 04 2004
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