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%I #16 Aug 05 2023 23:52:13
%S 1,1,1,0,1,1,0,1,1,1,0,1,4,1,1,0,1,26,14,1,1,0,1,236,509,49,1,1,0,1,
%T 2752,35839,10340,175,1,1,0,1,39208,4154652,5941404,222244,637,1,1,0,
%U 1,660032,718142257,7244337796,1081112575,4981531,2353,1,1,0,1,12818912,173201493539
%N Table read by descending antidiagonals. T(n,k) is the big Ramsey degree of k in w^n, where w is the first transfinite ordinal, omega.
%C T(n,k) is the least integer t such that, for all finite colorings of the k-subsets of w^n, there exists some S, an order-equivalent subset to w^n, where that coloring restricted to the k-subsets of S outputs at most t colors.
%C By Ramsey's theorem, the first row T(1,k)=1 for all k.
%C The second row T(2,k) coincides with A000311.
%C The second column T(n,2) coincides with A079309.
%D Dragan Mašulovic and Branislav Šobot, Countable ordinals and big Ramsey degrees, Combinatorica, 41 (2021), 425-446.
%D Alexander S. Kechris, Vladimir G. Pestov, and Stevo Todorčević, Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups, Geometric & Functional Analysis, 15 (2005), 106-189.
%H Joanna Boyland, William Gasarch, Nathan Hurtig, and Robert Rust, <a href="https://arxiv.org/abs/2305.07192">Big Ramsey Degrees of Countable Ordinals</a>, arXiv:2305.07192 [math.CO], 2023.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OrdinalNumber.html">Ordinal Number</a>.
%F T(n,k) = Sum_{p=0..n*k} P(p,n,k), where for n >= 2 and k >= 1,
%F P(0,n,k) = 0, and for p >= 1,
%F P(p,n,k) = Sum_{j=1..k} Sum_{0..p-1} binomial(p-1,i)*P(i,n-1,j)*P(p-1-i,n,k-j).
%e The data is organized in a table beginning with row n = 0 and column k = 0. The data is read by descending antidiagonals. T(2,3)=26.
%e The table T(n,k) begins:
%e [n/k] 0 1 2 3 4 5 ...
%e --------------------------------------------------------------------
%e [0] 1, 1, 0, 0, 0, 0, ...
%e [1] 1, 1, 1, 1, 1, 1, ...
%e [2] 1, 1, 4, 26, 236, 2572, ...
%e [3] 1, 1, 14, 509, 35839, 4154652, ...
%e [4] 1, 1, 49, 10340, ...
%e [5] 1, 1, 175, 222244, ...
%e [6] 1, 1, 637, ...
%o (Haskell)
%o pp p n k
%o | n == 0 && k >= 2 = 0
%o | k == 0 && p == 0 = 1
%o | k == 0 && p >= 1 = 0
%o | n == 0 && k == 1 && p == 0 = 1
%o | n == 0 && k == 1 && p >= 1 = 0
%o | n == 1 && k >= 1 && k == p = 1
%o | n == 1 && k >= 1 && k /= p = 0
%o | n >= 2 && k >= 1 = sum [binom (p-1) i * pp i (n-1) j * pp (p-1-i) n (k-j) | i <- [0..p-1], j <- [1..k]]
%o binom n 0 = 1
%o binom 0 k = 0
%o binom n k = binom (n-1) (k-1) * n `div` k
%o a364026 n k =
%o sum [pp p n k | p <- [0..n*k]]
%Y T(2,k) is A000311. T(n,2) is A079309.
%K nonn,tabl
%O 0,13
%A _Nathan Hurtig_, Jul 01 2023