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A364026 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.

%I A364026 #16 Aug 05 2023 23:52:13
%S A364026 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 A364026 2752,35839,10340,175,1,1,0,1,39208,4154652,5941404,222244,637,1,1,0,
%U A364026 1,660032,718142257,7244337796,1081112575,4981531,2353,1,1,0,1,12818912,173201493539
%N A364026 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 A364026 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 A364026 By Ramsey's theorem, the first row T(1,k)=1 for all k.
%C A364026 The second row T(2,k) coincides with A000311.
%C A364026 The second column T(n,2) coincides with A079309.
%D A364026 Dragan Mašulovic and Branislav Šobot, Countable ordinals and big Ramsey degrees, Combinatorica, 41 (2021), 425-446.
%D A364026 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 A364026 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 A364026 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OrdinalNumber.html">Ordinal Number</a>.
%F A364026 T(n,k) = Sum_{p=0..n*k} P(p,n,k), where for n >= 2 and k >= 1,
%F A364026 P(0,n,k) = 0, and for p >= 1,
%F A364026 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 A364026 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 A364026 The table T(n,k) begins:
%e A364026 [n/k]   0   1      2        3       4         5   ...
%e A364026 --------------------------------------------------------------------
%e A364026 [0]     1,  1,     0,       0,      0,        0,  ...
%e A364026 [1]     1,  1,     1,       1,      1,        1,  ...
%e A364026 [2]     1,  1,     4,      26,    236,     2572,  ...
%e A364026 [3]     1,  1,    14,     509,  35839,  4154652,  ...
%e A364026 [4]     1,  1,    49,   10340,  ...
%e A364026 [5]     1,  1,   175,  222244,  ...
%e A364026 [6]     1,  1,   637,  ...
%o A364026 (Haskell)
%o A364026 pp p n k
%o A364026   | n == 0 && k >= 2 = 0
%o A364026   | k == 0 && p == 0 = 1
%o A364026   | k == 0 && p >= 1 = 0
%o A364026   | n == 0 && k == 1 && p == 0 = 1
%o A364026   | n == 0 && k == 1 && p >= 1 = 0
%o A364026   | n == 1 && k >= 1 && k == p = 1
%o A364026   | n == 1 && k >= 1 && k /= p = 0
%o A364026   | 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 A364026 binom n 0 = 1
%o A364026 binom 0 k = 0
%o A364026 binom n k = binom (n-1) (k-1) * n `div` k
%o A364026 a364026 n k =
%o A364026   sum [pp p n k | p <- [0..n*k]]
%Y A364026 T(2,k) is A000311. T(n,2) is A079309.
%K A364026 nonn,tabl
%O A364026 0,13
%A A364026 _Nathan Hurtig_, Jul 01 2023