This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A232206 #169 Dec 20 2023 15:07:59
%S A232206 1,1,1,1,3,2,1,5,8,3,1,8,22,20,6,1,11,46,73,49,11,1,15,87,206,233,119,
%T A232206 23,1,19,147,485,807,689,288,46,1,24,236,1021,2320,2891,1988,696,98,1,
%U A232206 29,356,1960,5795,9800,9737,5561,1681,207,1,35,520,3525,13088,28586,38216,31350,15322,4062,451,1,41,730,5989,27224,74280,127465,139901,97552,41558,9821,983
%N A232206 Triangle read by rows: T(n,k) is the number of non-equivalent regular polygons with n+1 edges, one of which is rooted, which are dissected by non-intersecting diagonals into k regions, such that two such polygons are identified up to reflection along the rooted edge and twisting along the diagonals that does not affect the root edge (for 1 <= k <= n-1 and n >= 2).
%C A232206 From _Petros Hadjicostas_, Jan 20 2018: (Start)
%C A232206 According to Devadoss and Read (2001), the associahedron or Stasheff polytope K_n is a convex polytope of dim n-2 with co-dimension k faces corresponding to using k non-intersecting diagonals on a regular (n+1)-gon. If we redraw the picture of the dissected regular (n+1)-gon so that "every region of the dissection becomes a regular polygon without diagonals", then the resulting object is called a cluster, "where each regular polygon of the cluster is called a cell".
%C A232206 Two regular polygons G_1 and G_2 are of the same "dimension if their corresponding faces [on K_n] are of the same dimension"; of the same "type if their corresponding faces [on K_n] are the same polytope"; and of the same "class if they are identified under the actions of D_n [= dihedral group of order 2*n] and twisting [along the diagonals]".
%C A232206 In order to count how many faces of K_n belong to a given class (as defined above), Devadoss and Read (2001) follow the methodology of Read (1974, 1978) and first transform the regular (n+1)-gons (corresponding to the faces of K_n) into clusters, as mentioned above. Then they enumerate three kinds of clusters: those rooted at an outside edge (A-clusters); those rooted at a cell (B-clusters); and those rooted at an inside cell (C-clusters). In each one of these three kinds of clusters, two clusters are identified up to twisting along an edge (inside or outside depending on the case). Thus, the enumeration has to take twisting into account.
%C A232206 The g.f.s of these three kinds of clusters (A, B, and C) are derived, and these three g.f.s are used to derive the g.f. of the number of different classes of K_n of co-dimension k. See Section 3 in Devadoss and Read (2001).
%C A232206 The current triangular array, T(n,k), is the number of non-equivalent A-clusters (i.e., those rooted at an outside edge) having k cells and n outside edges, not counting the root edge, for 1 <= k <= n-1 and n>=2. Two such A-clusters are equivalent up to twisting the rooted edge or an inside edge separating two cells (as long as the inside twisting does not affect the rooted edge).
%C A232206 Read (1974, 1978) calls the A-clusters "out-rooted clusters". The difference between out-rooted clusters (= A-clusters) in Devadoss and Read (2001) and those in Read (1978) is that, in the first paper, two A-clusters are considered equivalent if one can be obtained from the other by twisting the rooted edge or an inside edge (separating two cells), while in the second paper twisting is not allowed.
%C A232206 If twisting is not allowed, then the number of (non-equivalent) A-clusters having k cells and n outside edges, not counting the root edge, is given by sequence A033282(n+1, k-1) = (1/k)*binomial(n-2,k-1)*binomial(n+k-1,k-1) for 1 <= k <= n-1 and n>=2. This is also the number of faces of K_n of co-dimension k-1. See Table 3 in Devadoss and Read (2001) and Table 1 in Read (1978). Unfortunately, in Table 1 in Read (1978), the label of each column is the number of outside edges including the rooted edge (i.e., n+1 rather than n).
%C A232206 Define T(n,k) = 0 for 0 <= n <= k, and T(n=1, k=0) = 1. Let also C(s,t) = Sum_{n>=0, k>=0} T(n,k)*s^n*t^k. (On p. 78 of their paper, Devadoss and Read (2001) use the notation a_{m,n} for T(n,m) and the notation A(x,y) for C(y,x), i.e., A(x,y) = Sum_{m>=0, n>=0} a_{m,n}*x^m*y^n in their paper.)
%C A232206 On p. 79, Eq. (3.2), of their paper, they prove that C(s,t) = s + (t/2)*(C(s,t)^2/(1-C(s,t)) + (1 + C(s,t))*C(s^2, t^2)/(1-C(s^2, t^2))). Unfortunately, the factor t in the previous formula is left out (i.e., it is a typo), and the same typo is reproduced in Schöbel and Veselov (2014, 2015).
%C A232206 We have C(s,t) = s+ t*s^2 + (t^2 + t)*s^3 + (2*t^3 + 3*t^2 + t)*s^4 + (3*t^4 + 8*t^3 + 5*t^2 + t)*s^5 + ...
%C A232206 Note that Sum_{0 <= k <= n-1} T(n,k) = A032132(n) for n>=1; T(n, k=1) = 1 for n>=2; T(k+1,k) = A001190(k+1) for k>=1, which are the Wedderburn-Etherington numbers; and T(n, k=2) = A024206(n-1) for n>=2.
%C A232206 According to Schöbel and Veselov (2014, 2015), for n>=2 and 0 <= p <= n-1, T(n+1, n-p) = A295380(n,p) is the number of inequivalent canonical forms for separation coordinates of the hypersphere S^n with p independent continuous parameters.
%C A232206 (End)
%D A232206 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 54-55 and 74-75.
%H A232206 Andrew Howroyd, <a href="/A232206/b232206.txt">Table of n, a(n) for n = 2..1276</a>
%H A232206 S. L. Devadoss and R. C. Read, <a href="https://doi.org/10.1007/PL00001293">Cellular structures determined by polygons and trees</a>, Ann. Combin., 5 (2001), 71-98.
%H A232206 Ronald C. Read, <a href="/A232206/a232206.pdf">On general dissections of a polygon</a>, Preprint (1974).
%H A232206 Ronald C. Read, <a href="http://dx.doi.org/10.1007/BF03031688">On general dissections of a polygon</a>, Aequat. Math., 18 (1978), 370-388.
%H A232206 K. Schöbel and A. Veselov, <a href="https://arxiv.org/abs/1307.6132">Separation coordinates, moduli spaces and Stasheff polytopes</a>, arXiv:1307.6132 [math.DG], 2013-2014.
%H A232206 K. Schöbel and A. Veselov, <a href="https://doi.org/10.1007/s00220-015-2332-x">Separation coordinates, moduli spaces and Stasheff polytopes</a>, Commun. Math. Phys., 337 (2015), 1255-1274.
%H A232206 Wikipedia, <a href="https://en.wikipedia.org/wiki/Associahedron">Associahedron</a>.
%F A232206 From _Petros Hadjicostas_, Jan 20 2018: (Start)
%F A232206 G.f.: If C(s,t) = Sum_{n>=0, k>=0} T(n,k)*s^n*t^k (with T(n=1,k=0) = 1), then C(s,t) = s + (t/2)*(C(s,t)^2/(1-C(s,t)) + (1 + C(s,t))*C(s^2, t^2)/(1-C(s^2, t^2))).
%F A232206 Note that t*C(s,t) = Sum_{k>=1} T(k,k-1)*(s*t)^k + Sum_{k>=2} (s*t)^k*Sum_{n >= k+1} T(n,k-1)*s^{n-k}. If we let w = s*t and D(s,w) = (w/s)*C(s,w/s), then the above functional equation implies that E(w): = lim_{s -> 0} D(s,w) = Sum_{k>=1} T(k,k-1)*w^k satisfies E(w) = w + (1/2)*(E(w)^2 + E(w^2)), which is the g.f. for the Wedderburn-Etherington numbers in sequence A001190. Thus, T(k,k-1) = A001190(k) for k>=1.
%F A232206 Expansion w.r.t to t: C(s,t) = s + t*s^2/(1-s) + t^2*s^3*(1+s-s^2)/((1+s)*(1-s)^3) + t^3*s^4*(2+2*s-2*s^3+s^4)/((1+s)^2*(1-s)^5) + t^4*s^5*(3+8*s+2*s^2-2*s^3-2*s^4+3*s^5-s^6)/((1+s)^3*(1-s)^7) + t^5*s^6*(6+19*s+30*s^2+15*s^3+23*s^4-4*s^5+2*s^6-4*s^7+6*s^8-4*s^9+s^10)/((1+s^2)*(1+s)^4*(1-s)^9) + ... (It can be proven using a symbolic computation package by manipulating the above functional equation by Devadoss and Read.)
%F A232206 This means that, in the above expansion for C(s,t), the coefficient of t is the g.f. of the first column of the triangle below, the coefficient of t^2 is the g.f. of the second column of the triangle below, and so on.
%F A232206 (End)
%e A232206 From _Petros Hadjicostas_, Jan 20 2018: (Start)
%e A232206 According to Devadoss and Read (2015), triangle T(n,k) begins as follows:
%e A232206 (n=2) 1,
%e A232206 (n=3) 1, 1,
%e A232206 (n=4) 1, 3, 2,
%e A232206 (n=5) 1, 5, 8, 3,
%e A232206 (n=6) 1, 8, 22, 20, 6,
%e A232206 (n=7) 1, 11, 46, 73, 49, 11,
%e A232206 (n=8) 1, 15, 87, 206, 233, 119, 23,
%e A232206 (n=9) 1, 19, 147, 485, 807, 689, 288, 46,
%e A232206 (n=10) 1, 24, 236, 1021, 2320, 2891, 1988, 696, 98,
%e A232206 (n=11) 1, 29, 356, 1960, 5795, 9800, 9737, 5561, 1681, 207,
%e A232206 ...
%e A232206 Geometrical example for n=5: If no twisting is allowed, the number of regular (n+1)-gons (= hexagons) with a rooted edge and dissected into k regions by non-intersecting diagonals is given by A033282(n+1, k-1) = A033282(6, k-1): 1, 9, 21, and 14 for k = 1, 2, 3, 4, respectively.
%e A232206 Recall that, according to Devadoss and Read (2001), two regular (unrooted) polygons G_1 and G_2 are of the same "class" if they are identified under the actions of D_n (= dihedral group of order 2*n) and twisting along the diagonals. To avoid confusion, we call two such unrooted equivalent polygons as being of the same DT-class (D for "dihedral" and T for "twisting"). According to Table 5 (p. 92) in Devadoss and Read (2001), the numbers of DT-classes for regular hexagons dissected into k regions (by k-1 non-intersecting diagonals) are 1, 2, 3, 2 for k = 1, 2, 3, 4, respectively. When an edge is rooted, each one of these DT-classes is subdivided into subclasses.
%e A232206 Call the regular hexagons ABCDEF with FA being the rooted edge (base).
%e A232206 For k=1, we have T(5,1) = 1 rooted regular hexagon with no intersecting diagonals.
%e A232206 For k=2, we have T(5,2) = 5 equivalent rooted regular hexagons with 1 diagonal. The equivalence classes are as follows according to the single dissecting diagonal:
%e A232206 Class 1a: AE, FB; Class 1b: AC, FD; Class 1c: BD, EC; (DT-class 1 has 6 hexagons)
%e A232206 Class 2a: AD, FC; Class 2b: BE; (DT-class 2 has 3 hexagons).
%e A232206 Note that 6 + 3 = 9 = A033282(5+1, 2-1).
%e A232206 For k=3, we have T(5,3) = 8 equivalent rooted regular hexagons with 2 non-intersecting diagonals. The equivalence classes are as follows according to the two diagonals:
%e A232206 Class 1a: (AC, AE), (FB, FD), (BD, BF), (EA, EC);
%e A232206 Class 1b: (DB, DF), (CE, CA); (DT-class 1 has 6 hexagons)
%e A232206 Class 2a: (DB, EA), (CE, BF);
%e A232206 Class 2b: (DF, CA); (DT-class 2 has 3 hexagons)
%e A232206 Class 3a: (DA, EA), (FC, FB), (EA, EB), (BE, BF);
%e A232206 Class 3b: (AC, AD), (FC, FD), (DA, DB), (CE, CF);
%e A232206 Class 3c: (BD, BE), (EC, EB);
%e A232206 Class 3d: (CF, CA), (DA, DF); (DT-class 3 has 12 hexagons)
%e A232206 Note that 6 + 3 + 12 = 21 = A033282(5+1, 3-1).
%e A232206 For k=4, we have T(5,4) = 3 equivalent rooted regular hexagons with 3 non-intersecting diagonals. The equivalence classes are as follows according to the three diagonals:
%e A232206 Class 1: (EA, AC, CE), (BD, DF, FB); (DT-class 1 has 2 hexagons)
%e A232206 Class 2a: (DF, DA, AC), (DF, FC, CA), (CE, CF, CA), (DF, DA, DB);
%e A232206 Class 2b: (CE, BE, BF), (BD, BE, BF), (EC, EB, EA), (DB, EB, EA), (EC, CF, FB), (DB, DA, AE), (FD, FC, FB), (AE, AD, AC); (DT class 2 has 12 hexagons)
%e A232206 Note that 2 + 12 = 14 = A033282(5+1, 4-1).
%e A232206 Recall that two rooted hexagons are equivalent iff they are a reflection of each other along the rooted edge or one can be obtained from the other by twisting a diagonal as long as the twisting does not affect the rooted edge.
%e A232206 The case k = 4 = n-1 above is related to the triangulation of a convex polygon and the Wedderburn-Etherington commutative bracketing problem that appear in Comtet (1974, pp. 54-55). Devadoss and Read (2001, p. 89) claim that T(n,k) is the number of possible ways of using k-1 pairs of brackets on n commutative variables, but it is not clear how each one of the above hexagons (from k=1 to k=3) can be transformed into some kind of a generalized commutative bracketing problem.
%e A232206 (End)
%t A232206 rows = 12; A[_, _] = 0;
%t A232206 Do[A[s_, t_] = Series[s+(t/2)(A[s, t]^2/(1-A[s, t])+(1+A[s, t]) A[s^2, t^2] / (1-A[s^2, t^2])), {s, 0, rows+1}, {t, 0, rows+1}] // Normal, {rows+1}];
%t A232206 Rest[CoefficientList[#, t]]& /@ Drop[CoefficientList[A[s, t], s], 2] // Flatten (* _Jean-François Alcover_, Sep 02 2018 *)
%o A232206 (SageMath)
%o A232206 #This is a crude Sage program on how to derive the g.f. of any column in the triangle given that you have correctly derived the g.f.s of the previous columns. Suppose you have the expansion for C(s,t) w.r.t. t up to the power of 3 and you want to get the coefficient of t^4.
%o A232206 s,t=var('s,t')
%o A232206 C(s,t) = s + t*s^2/(1-s) + t^2*s^3*(1+s-s^2)/((1+s)*(1-s)^3) + t^3*s^4*(2+2*s-2*s^3+s^4)/((1+s)^2*(1-s)^5)
%o A232206 B(s,t)=s+(t/2)*(C(s,t)^2/(1-C(s,t))+(1+C(s,t))*C(s^2,t^2)/(1-C(s^2,t^2)))-C(s,t)
%o A232206 factor(taylor(B(s,t),t,0,4)/t^4)
%o A232206 # _Petros Hadjicostas_, Jan 21 2018
%o A232206 (PARI)
%o A232206 BIKxy(p)={(1/(1-p) + (1+p)/subst(subst(1-p, x, x^2), y, y^2))/2}
%o A232206 A(n)={my(p=x); for(i=2, n+1, p+=x^i*y*polcoeff(BIKxy(p + O(x*x^i)), i)); [Vecrev(q/y) | q<-Vecrev((p-x)/x^2)]}
%o A232206 { my(T=A(10)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Aug 31 2018
%Y A232206 Cf. A001190 (main diagonal), A024206 (second column), A032132 (row sums), A295380 (mirror image).
%K A232206 nonn,tabl
%O A232206 2,5
%A A232206 _N. J. A. Sloane_, Nov 21 2013
%E A232206 Name edited by _Petros Hadjicostas_, Jan 20 2018
%E A232206 Offset corrected by _Andrew Howroyd_, Aug 31 2018