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A097877
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Triangle read by rows: T(n,k) is the number of Dyck n-paths with k large components, 0 <= k <= n/2.
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0
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1, 1, 1, 1, 1, 4, 1, 12, 1, 1, 34, 7, 1, 98, 32, 1, 1, 294, 124, 10, 1, 919, 448, 61, 1, 1, 2974, 1576, 298, 13, 1, 9891, 5510, 1294, 99, 1, 1, 33604, 19322, 5260, 583, 16, 1, 116103, 68206, 20595, 2960, 146, 1, 1, 406614, 242602, 78954, 13704, 1006, 19, 1, 1440025
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| A prime Dyck path is one with exactly one return to ground level. Every nonempty Dyck path decomposes uniquely as a concatenation of prime Dyck paths, called its components. For example, UUDDUD has 2 components: UUDD and UD, of semilength 2 and 1 respectively. A large component is one of semilength >= 2.
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FORMULA
| G.f.: 2/(1 + t*(1 - 4*z)^(1/2) + (1 - 2*z)(1-t)) = Sum_{n>=0, k>=0} T(n, k) z^n t^k satisfies (1-z)*G = 1 + z*t*(CatalanGF[z]-1)*G. The gf for Dyck paths (A000108) with z marking semilength is CatalanGF[z]:=(1 - sqrt[1 - 4*z])/(2*z). Hence the gf for prime Dyck paths is z*CatalanGF[z] and the gf for non-UD prime Dyck paths is S(z):= z*CatalanGF[z]-z. For fixed k, the gf for (T(n, k))_{n>=0} is S(z)^k/(1-z)^(k+1). This is clear because 1/(1-z) is the gf for all-UD Dyck paths (including the empty one) and a Dyck path with k large components is a product (uniquely) of an all-UD, a non-UD prime, an all-UD, a non-UD prime, ... with k non-UD primes and k+1 all-UDs.
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EXAMPLE
| Example: Table begins
\ k 0, 1, 2, ...
n
0 | 1
1 | 1
2 | 1, 1
3 | 1, 4,
4 | 1, 12, 1
5 | 1, 34, 7
6 | 1, 98, 32, 1
7 | 1, 294, 124, 10
8 | 1, 919, 448, 61, 1
T(3,1)=4 because each of the 5 Dyck paths of semilength 3 has one
large component except for UDUDUD, which has none.
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CROSSREFS
| The Fine distribution (A065600) counts Dyck paths by number of small components (= number of low peaks).
Sequence in context: A109088 A060923 A143952 * A019304 A072869 A064279
Adjacent sequences: A097874 A097875 A097876 * A097878 A097879 A097880
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KEYWORD
| nonn,tabl
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AUTHOR
| David Callan (callan(AT)stat.wisc.edu), Sep 21 2004
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