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A259179
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Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma.
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17
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1, 2, 2, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 0, 4, 0, 1, 3, 0, 2, 0, 2, 3, 0, 1, 4, 0, 2, 0, 3, 0, 3, 0, 1, 1, 4, 0, 2, 0, 4, 0, 3, 0, 1, 2, 0, 4, 0, 2, 0, 0, 5, 0, 3, 0, 1, 3, 0, 4, 0, 2, 0, 1, 0, 5, 0, 2, 1, 0, 1, 4, 0, 4, 0, 2, 0, 2, 0, 5, 0, 3, 0, 0, 0, 1, 5, 0, 2, 2, 0, 2, 0, 3, 0, 5, 0, 3, 0, 1, 0, 0, 6
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OFFSET
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1,2
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COMMENTS
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Since the diagram of the symmetric representation of sigma is also the top view of the stepped pyramid described in A245092, and the diagram is also the top view of the staircase described in A244580, so we have that a(n) is also the height difference (or length of the vertical line segment) at the point (n,n) in the main diagonal of the mentioned structures.
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LINKS
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EXAMPLE
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Illustration of initial terms:
--------------------------------------------------------
Diagram with 15 Dyck paths
n A000203(n) a(n) to evaluate a(1)..a(10)
--------------------------------------------------------
. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1 1 1 |_| | | | | | | | | | | | | | |
2 3 2 |_ _|_| | | | | | | | | | | | |
3 4 2 |_ _| _|_| | | | | | | | | | |
4 7 0 |_ _ _| _|_| | | | | | | | |
5 6 2 |_ _ _| _| _ _|_| | | | | | |
6 12 1 |_ _ _ _| _| | _ _|_| | | | |
7 8 3 |_ _ _ _| |_ _|_| _ _|_| | |
8 15 0 |_ _ _ _ _| _| | _ _ _|_|
9 13 3 |_ _ _ _ _| | _|_| |
10 18 0 |_ _ _ _ _ _| _ _| _|
. |_ _ _ _ _ _| | _| _|
. |_ _ _ _ _ _ _| |_ _|
. |_ _ _ _ _ _ _| |
. |_ _ _ _ _ _ _ _|
. |_ _ _ _ _ _ _ _|
.
For n = 3 there are two Dyck paths that contain the point (3,3) so a(3) = 2.
For n = 4 there are no Dyck paths that contain the point (4,4) so a(4) = 0.
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MATHEMATICA
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a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a259179[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; t] (* a(1..n) *)
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CROSSREFS
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Cf. A000203, A024916, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A249351, A256533, A259179, A262626, A279286, A280919.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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