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%I #66 Jan 31 2018 04:02:30
%S 1,1,1,2,3,1,3,8,5,1,6,20,22,8,1,11,49,73,46,11,1,23,119,233,206,87,
%T 15,1,46,288,689,807,485,147,19,1,98,696,1988,2891,2320,1021,236,24,1,
%U 207,1681,5561,9737,9800,5795,1960,356,29,1,451,4062,15322,31350,38216,28586,13088,3525,520,35,1,983,9821,41558,97552,139901,127465,74280,27224,5989,730,41,1
%N Number of canonical forms for separation coordinates on hyperspheres S_n, ordered by increasing number of independent continuous parameters.
%C Table 1 of the Schöbel and Veselov paper with initial 1 added. Reverse of Table 2 of the Devadoss and Read paper.
%C Apparently A032132 contains the row sums.
%C From _Petros Hadjicostas_, Jan 28 2018: (Start)
%C In this triangle, which is read by rows, for 0 <= k <= n-1 and n>=1, let T(n,k) be the number of inequivalent canonical forms for separation coordinates of the hypersphere S^n with k independent continuous parameters. It is the mirror image of sequence A232206, that is, T(n, k) = A232206(n+1, n-k) for 0 <= k <= n-1 and n>=1. (Triangular array A232206(N, K) is defined for N >= 2 and 1 <= K <= N-1.)
%C If B(x,y) = Sum_{n,k>=0} T(n,k)*x^n*y^k (with T(0,0) = 1, T(0,k) = 0 for k>=1, and T(n,k) = 0 for 1 <= n <= k), then B(x,y) = 1 + (x/2)*(B(x,y)^2/(1-x*y*B(x,y)) + (1 + x*y*B(x,y))*B(x^2,y^2)/(1-x^2*y^2*B(x^2,y^2))). This can be derived from the bivariate g.f. of A232206. See the comments for that sequence.
%C Let S(n) := Sum_{k>=0} T(n,k). The g.f. of S(n) is B(x, y=1). If we let y=1 in the above functional equation, we get x*B(x,1) = x + (1/2)*((x*B(x,1))^2/(1-x*B(x,1)) + (1 + x*B(x,1))*x^2*B(x^2,1)/(1-x^2*B(x^2,1))). After some algebra, we get 2*x*B(x,1) = x + (1/2)(x*B(x,1)/(1-x*B(x,1)) + (x*B(x,1) + x^2*B(x^2,1))/(1-x^2*B(x,1))), i.e., 2*x*B(x,1) = x + BIK(x*B(x,1)), where we have the "BIK" (reversible, indistinct, unlabeled) transform of C. G. Bower. This proves that S(n) = A032132(n+1) for n>=0, which is Copeland's claim above.
%C Note that for the second column we have T(n,k=2) = A048739(n-2) for 2 <= n < = 10, but T(11,2) = 4062 <> 4059 = A048739(9). In any case, they have different g.f.s (see the formula section below).
%C (End)
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H S. Devadoss and R. C. Read, <a href="https://arxiv.org/abs/math/0008145">Cellular structures determined by polygons and trees</a>, arXiv/0008145 [math.CO], 2000.
%H 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 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], 2014.
%H 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.
%F From _Petros Hadjicostas_, Jan 28 2018: (Start)
%F G.f.: If B(x,y) = Sum_{n,k>=0} T(n,k)*x^n*y^k (with T(0,0) = 1, T(0,k) = 0 for k>=1, and T(n,k) = 0 for 1 <= n <= k), then B(x,y) = 1 + (x/2)*(B(x,y)^2/(1-x*y*B(x,y)) + (1 + x*y*B(x,y))*B(x^2,y^2)/(1-x^2*y^2*B(x^2,y^2))).
%F If c(N,K) = A232206(N,K) and C(x,y) = Sum_{N,K>=0} c(N,K)*x^N*y^K (with c(1,0) = 1 and c(N,K) = 0 for 0 <= N <= K), then C(x,y) = x*B(x*y, 1/y) and B(x,y) = C(x*y, 1/y)/(x*y).
%F Setting y=0 in the above functional equation, we get x*B(x,0) = x + (1/2)*((x*B(x,0))^2 + x^2*B(x^2,0)), which is the functional equation for the g.f. of the first column. This proves that T(n,k=0) = A001190(n+1) for n>=0 (assuming T(0,0) = 1).
%F The g.f. of the second column is B_1(x,0) = Sum_{n>=0} T(n,2)*x^n = lim_{y->0} (B(x,y)-B(x,0))/y, where B(x,0) = 1 + x + x^2 + ... is the g.f. of the first column. We get B_1(x,0) = x*B(x,0)*(B(x,0) - 1)/(1 - x*B(x,0)).
%F (End)
%e From _Petros Hadjicostas_, Jan 27 2018: (Start)
%e Triangle T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9
%e ----------------------------------------------------------------
%e (S^1) 1,
%e (S^2) 1, 1,
%e (S^3) 2, 3, 1,
%e (S^4) 3, 8, 5, 1,
%e (S^5) 6, 20, 22, 8, 1,
%e (S^6) 11, 49, 73, 46, 11, 1,
%e (S^7) 23, 119, 233, 206, 87, 15, 1,
%e (S^8) 46, 288, 689, 807, 485, 147, 19, 1,
%e (S^9) 98, 696, 1988, 2891, 2320, 1021, 236, 24, 1,
%e (S^10) 207, 1681, 5561, 9737, 9800, 5795, 1960, 356, 29, 1,
%e ...
%e (End)
%Y Cf. A001190, A024206, A032132, A232206.
%K nonn,tabl
%O 1,4
%A _Tom Copeland_, Nov 21 2017
%E Typo for T(11,3)=15322 corrected by _Petros Hadjicostas_, Jan 28 2018
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