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A245192 The number of Dyck paths p(m) for m<=n, as defined by the rows of A237593, that have common subpaths of positive length with the Dyck path p(n) for the symmetric representation of sigma(n).

%I A245192 #25 Jun 27 2025 00:46:45
%S A245192 0,1,1,1,1,2,1,2,1,2,1,2,1,2,3,2,1,2,1,2,1,2,2,3,1,2,3,3,1,2,1,2,1,2,
%T A245192 3,3,1,2,3,4,1,2,1,2,3,2,2,3,1,2,3,2,3,4,1,2,1,2,3,4,1,2,3,4,1,2,1,2,
%U A245192 3,4,2,3,1,2,3,4,5,2,1,2,1,2,3,4,1,2,3,4,1,2,1
%N A245192 The number of Dyck paths p(m) for m<=n, as defined by the rows of A237593, that have common subpaths of positive length with the Dyck path p(n) for the symmetric representation of sigma(n).
%C A245192 This sequence counts Dyck paths that have common stretches while sequence A244145 counts adjacent areas of the symmetric representation of sigma(). Their first three differences occur at positions 23, 47 and 53.
%C A245192 A244145 counts adjacent sections rather than common boundaries.
%C A245192 See A237270 for Mathematica function used here.
%H A245192 Hartmut F. W. Hoft, <a href="/A245192/a245192.pdf">Image of overlapping Dyck paths</a>
%e A245192 Path a(6) has two colors since it shares steps 5 and 6 with path a(5) which has a single color.
%e A245192 See also the link for a color image of paths.
%t A245192 (* path[n] computing the n-th Dyck path is defined in A237270 *)
%t A245192 (* coloredPathRange[] assigns the color of the first path sharing a line *)
%t A245192 (* colorLists[] computes the lists of colors in each path in the list *)
%t A245192 defaultPath[n_] := Module[{p=path[n]}, Transpose[{Transpose[{Most[p], Rest[p]}], Table[n, {Length[p]-1}]}]]
%t A245192 switchIf[x_,yList_] := Module[{pos=Position[Map[First, yList], First[x]]}, If[pos == {}, x, yList[[First[First[pos]]]]]]
%t A245192 nextColoredPath[p_,n_] := Module[{u=defaultPath[n], meet12, common1}, meet12 = Intersection[Map[First, p], Map[First, u]]; common1=Select[p, MemberQ[meet12, First[#]]&]; Map[switchIf[#, common1]&, u]]
%t A245192 coloredPathRange[n_] := FoldList[nextColoredPath, {{{{0,0}, {0,0}}, 0}}, Range[n]]
%t A245192 colorLists[pathList_] := Map[Union[Last[Transpose[#]]]&, pathList]
%t A245192 a[colors_] := Prepend[Map[Last[#] - First[#] + 1&, Rest[colors]], 0]
%t A245192 a[colorLists[coloredPathRange[90]]] (* computes the first 90 values *)
%Y A245192 Cf. A237270, A237593, A244145.
%K A245192 nonn
%O A245192 1,6
%A A245192 _Hartmut F. W. Hoft_, Jul 17 2014