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 A369605 #18 May 14 2025 13:26:53 %S A369605 1,1,1,1,1,1,1,1,1,1,3,2,3,1,1,1,1,1,3,5,11,19,36,37,41,24,18,6,4,1,1, %T A369605 1,1,1,3,5,17,44,158,493,1628,4670,12266,27043,51018,79042,103179, %U A369605 112219,105232,84045,59021,35533,19114,8769,3716,1311,468,130,47,10,5,1,1 %N A369605 Irregular triangle read by rows: T(n,k) is the number of inequivalent connected induced k-vertex subgraphs of the hypercube graph of dimension n >= 0, 1 <= k <= 2^n. %C A369605 Two subgraphs are equivalent if there is an automorphism of the hypercube graph that takes one to the other. %C A369605 Two isomorphic subgraphs may both be counted. For example, the path with 5 vertices is an induced subgraph of the 4-dimensional hypercube in two inequivalent ways: one that is contained in a 3-dimensional subcube and one that is not. This implies that T(4,5) > A369997(4,5). (In A369997, the subgraphs are counted up to isomorphism.) %C A369605 Also, number of free k-celled polycubes in n dimensions, whose width in any coordinate direction is at most 2. %C A369605 Also, number of k-celled polyominoes whose cells are subsets of the (n-1)-dimensional facets of the n-dimensional cross-polytope (or orthoplex). (See A049540.) %C A369605 A039754 is the corresponding sequence for not necessarily connected subgraphs. %F A369605 T(n,k) = A049540(k) for k <= n+1. %F A369605 T(n,k) = A039754(n,k) for k > 2^n-n. %e A369605 Triangle begins: %e A369605 1; %e A369605 1, 1; %e A369605 1, 1, 1, 1; %e A369605 1, 1, 1, 3, 2, 3, 1, 1; %e A369605 1, 1, 1, 3, 5, 11, 19, 36, 37, 41, 24, 18, 6, 4, 1, 1; %e A369605 ... %e A369605 There are T(3,4) = 3 inequivalent connected induced 4-vertex subgraphs of the 3-cube: four vertices of a 2-dimensional face or three vertices of a face together with a vertex from the opposite face, adjacent to either of two inequivalent vertices from the first face. %Y A369605 Cf. A049540 (main diagonal), A333333 (edge-induced subgraphs). %Y A369605 Different ways of counting induced subgraphs in the hypercube graph (totals or by number of vertices): %Y A369605 \ Subgraphs | All | Connected %Y A369605 Symmetries \ | | %Y A369605 --------------------------+-----------------+---------------- %Y A369605 None | A001146/ N/A | A290758/A369999 %Y A369605 Automorphisms of the cube | A000616/A039754 | A369606/A369605 %Y A369605 Isomorphism | A369996/A369995 | A369998/A369997 %Y A369605 (The N/A entry corresponds to rows 2^n of Pascal's triangle; A345135 comes close.) %K A369605 nonn,tabf %O A369605 0,11 %A A369605 _Pontus von Brömssen_, Jan 27 2024 %E A369605 Row 5 from _Pontus von Brömssen_, May 14 2025