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 A385437 #8 Jul 03 2025 16:39:26
%S A385437 1,2,4,1,1,10,20,9,1,1,18,92,146,80,16,1,1,35,355,1146,1492,850,220,
%T A385437 25,1,1,68,1336,7590,17831,19740,11052,3230,490,36,1,1,133,5026,47278,
%U A385437 181251,332039,320763,172788,53417,9520,952,49,1,1,262,19097,287126,1710016,4809728,7204912,6180858,3177106,1003940,196728,23660,1680,64,1
%N A385437 Triangle read by rows: T(n,k) is the number of proper vertex colorings of the n-complete bipartite graph with a perfect matching removed using exactly k interchangeable colors, for n >= 1 and 2 <= k <= 2n.
%C A385437 Permuting the colors does not change the coloring. T(n,k) is the number of ways to partition the vertex set of the n-complete bipartite graph with a perfect matching removed into k nonempty independent sets, for n >= 1 and 2 <= k <= 2n. T(n,2) = 1 for all n >= 1, corresponding to the partition into the two vertex sets. T(n,2n) = 1 for all n >= 1, corresponding to the partition where each vertex forms its own independent set.
%H A385437 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>.
%F A385437 T(n,k) = Sum[Binomial[n, s]*Sum[StirlingS2[s, j]*StirlingS2[s, k - n + s - j], {j, 0, k - n + s}], {s, 0, n}].
%e A385437 Triangle begins (n >= 1, k >= 2):
%e A385437 n = 1: [1]
%e A385437 n = 2: [2, 4, 1]
%e A385437 n = 3: [1, 10, 20, 9, 1]
%e A385437 n = 4: [1, 18, 92, 146, 80, 16, 1]
%e A385437 n = 5: [1, 35, 355, 1146, 1492, 850, 220, 25, 1]
%e A385437 n = 6: [1, 68, 1336, 7590, 17831, 19740, 11052, 3230, 490, 36, 1]
%e A385437 n = 7: [1, 133, 5026, 47278, 181251, 332039, 320763, 172788, 53417, 9520, 952, 49, 1]...
%e A385437 For n=2, k=3: T(2,3) = 4. The graph K_{2,2} - M has vertices {a_1, a_2, b_1, b_2} with edges {a_1,b_2}, {a_2,b_1}, {a_2,b_2}, {a_1,b_1} (assuming the perfect matching M = {{a_1,b_1}, {a_2,b_2}} is removed). The 4 ways to partition into 3 independent sets are: {{a_1},{a_2},{b_1,b_2}}, {{a_1},{b_1},{a_2,b_2}}, {{a_2},{b_2},{a_1,b_1}}, {{b_1},{b_2},{a_1,a_2}}.
%o A385437 (PARI) T(n, k) = sum(s=0, n, binomial(n, s)*sum(j=0, k - n + s, stirling(s, j, 2)*stirling(s, k - n + s - j, 2)));
%o A385437 for(n=1, 10, print(vector(2*n - 1, k, T(n, k + 1))))
%K A385437 nonn,tabf
%O A385437 1,2
%A A385437 _Julian Allagan_, Jun 28 2025