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 A360849 Array read by antidiagonals: T(m,n) is the number of (undirected) cycles in the complete bipartite graph K_{m,n}. 4
 0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 15, 6, 0, 0, 10, 42, 42, 10, 0, 0, 15, 90, 204, 90, 15, 0, 0, 21, 165, 660, 660, 165, 21, 0, 0, 28, 273, 1650, 3940, 1650, 273, 28, 0, 0, 36, 420, 3486, 15390, 15390, 3486, 420, 36, 0, 0, 45, 612, 6552, 45150, 113865, 45150, 6552, 612, 45, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Also, T(m,n) is the number of chordless cycles of length >= 4 in the m X n rook graph. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 Eric Weisstein's World of Mathematics, Chordless Cycle. Eric Weisstein's World of Mathematics, Complete Bipartite Graph. Eric Weisstein's World of Mathematics, Graph Cycle. Eric Weisstein's World of Mathematics, Rook Graph. FORMULA T(m,n) = Sum_{j=2..min(m,n)} binomial(m,j)*binomial(n,j)*j!*(j-1)!/2. T(m,n) = T(n,m). EXAMPLE Array begins: ======================================================== m\n| 1 2 3 4 5 6 7 8 ... ---+---------------------------------------------------- 1 | 0 0 0 0 0 0 0 0 ... 2 | 0 1 3 6 10 15 21 28 ... 3 | 0 3 15 42 90 165 273 420 ... 4 | 0 6 42 204 660 1650 3486 6552 ... 5 | 0 10 90 660 3940 15390 45150 109480 ... 6 | 0 15 165 1650 15390 113865 526155 1776180 ... 7 | 0 21 273 3486 45150 526155 4662231 24864588 ... 8 | 0 28 420 6552 109480 1776180 24864588 256485040 ... ... Lower half of array as triangle T(n,k) for 1 <= k <= n begins: 0; 0, 1; 0, 3, 15; 0, 6, 42, 204; 0, 10, 90, 660, 3940; 0, 15, 165, 1650, 15390, 113865; 0, 21, 273, 3486, 45150, 526155, 4662231; ... PROG (PARI) T(m, n) = sum(j=2, min(m, n), binomial(m, j)*binomial(n, j)*j!*(j-1)!/2) CROSSREFS Rows 1..3 are A000004, A000217(n-1), A059270(n-1). Main diagonal is A070968. Cf. A269562, A286418, A360850 (paths), A360853. Sequence in context: A299554 A300175 A129533 * A155999 A338034 A299904 Adjacent sequences: A360846 A360847 A360848 * A360850 A360851 A360852 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Feb 23 2023 STATUS approved

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Last modified August 15 01:27 EDT 2024. Contains 375171 sequences. (Running on oeis4.)