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Search: id:A098093
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| A098093 |
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Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k ladders. A string of consecutive up steps U_1, U_2, ..., U_m and their matching down steps D_1, D_2, ..., D_m are said to form a ladder if (i) D_1, D_2, ..., D_m are consecutive steps and (ii) the sequence of pairs (U_j, D_j) (j=1,2,...,m) is maximal. For example, in the path (UU)[U]H[D]H(DD), where U=(1,1), H=(1,0), D=(1,-1), we have 2 ladders, shown between parentheses and square brackets, respectively. |
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1
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| 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 13, 3, 1, 22, 14, 1, 34, 46, 1, 1, 50, 118, 16, 1, 70, 264, 88, 1, 95, 530, 343, 9, 1, 125, 986, 1066, 105, 1, 161, 1722, 2857, 630, 2, 1, 203, 2863, 6841, 2751, 76, 1, 252, 4564, 15028, 9746, 781, 1, 308, 7028, 30778, 29778, 4909, 30
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OFFSET
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0,7
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COMMENT
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Row sums yield the RNA secondary structure numbers (A004148). Column 1 is A002623.
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REFERENCES
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I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26, 1979, 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
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LINKS
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M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.
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FORMULA
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G.f.=G=G(t, z) satisfies G=1+zG+tz^2*G(G-1)/(1-z^2+tz^2).
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EXAMPLE
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Triangle starts:
1;
1;
1;
1,1;
1,3;
1,7;
1,13,3;
1,22,14;
1,34,46,1;
Apparently, rows 5n+1 and 5n+2 have 2n+1 terms and rows 5n+3,5n+4 and 5n+5 have 2n+2 terms.
T(6,2)=3 because we have (U)H(D)[U]H[D], (U)H[U]H[D](D) and (U)[U]H[D]H(D), the two ladders being shown between parentheses and square brackets, respectively.
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CROSSREFS
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Cf. A004148, A002623.
Sequence in context: A077202 A086665 A050521 this_sequence A160627 A114712 A089741
Adjacent sequences: A098090 A098091 A098092 this_sequence A098094 A098095 A098096
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 14 2004
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