A048193
Number of chordal graphs (or triangulated graphs) on n vertices.
Original entry on oeis.org
1, 2, 4, 10, 27, 94, 393, 2119, 14524, 126758, 1392387, 19109099, 326005775, 6905776799, 181945055235
Offset: 1
Cf.
A048192 (connected chordal graphs).
Cf.
A287427 (disconnected chordal graphs).
A179534
Number of labeled split graphs on n vertices.
Original entry on oeis.org
1, 2, 8, 58, 632, 9654, 202484, 5843954, 233064944, 12916716526, 998745087980, 108135391731690, 16434082400952296, 3513344943520006118, 1058030578581541945316, 449389062270642095128546, 269419653009366144571801568, 228157953744571034350576205790
Offset: 1
- V. Bina, Enumeration of Labeled Split Graphs and Counts of Important Superclasses. In Adacher L., Flamini, M., Leo, G., Nicosia, G., Pacifici, A., Picialli, V. (Eds.). CTW 2011, Villa Mondragone, Frascati, pp. 72-75 (2011).
- Andrew Howroyd, Table of n, a(n) for n = 1..100
- E. A. Bender, L. B. Richmond, and N. C. Wormald, Almost all chordal graphs split, J. Austral. Math. Soc. (Series A), 38 (1985), 214-221.
- V. Bina, Multidimensional probability distributions: Structure and learning, PHD Thesis. Fac. of Management, University of Economics in Prague (2011).
- Venkatesan Guruswami, Enumerative aspects of certain subclasses of perfect graphs, Discrete Mathematics 205 (1999), 97-117.
- J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2018-2019.
- J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.
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A179534 := proc(n) local a,k; a := 1 ; for k from 2 to n do a := a+binomial(n,k)*( (2^k-1)^(n-k) -add(j*k*binomial(n-k,j)*(2^(k-1)-1)^(n-k-j)/(j+1), j=1..n-k) ) end do: a ; end proc: # R. J. Mathar, Jun 21 2011
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a[n_] := 1 + Sum[Binomial[n,k]*((2^k-1)^(n-k) - Sum[j*k*Binomial[n-k,j]*(2^(k-1)-1)^(n-k-j)/(j+1), {j,1,n-k}]), {k,2, n}]; Array[a, 20] (* Stefano Spezia, Oct 29 2018 *)
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b(n) = {sum(k=0, n, binomial(n, k)*2^(k*(n-k)))} \\ A047863(n)
a(n) = b(n) - n*b(n-1) \\ Andrew Howroyd, Jun 06 2021
a(12)-a(16) corrected and terms a(17) and beyond from
Andrew Howroyd, Jun 06 2021
A347699
Triangle read by rows: For n >= 1, 0 <= k <= n-1, T(n,k) = 0 if k=0, otherwise the number of inequivalent k X (n-k) 0,1 matrices having at least one 1 in each column.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 9, 23, 17, 5
Offset: 1
Triangle begins:
1;
1, 1;
1, 1, 2;
1, 1, 4, 3;
1, 1, 6, 9, 4;
1, 1, 9, 23, 17, 5;
...
Original entry on oeis.org
1, 2, 5, 18, 105, 946, 12589, 240482, 6526289, 250119330, 13512676053, 1027978959346, 110155994874553, 16631509898085074, 3540687511804739837, 1063409634646294541250, 450894476653951603096737
Offset: 0
A368761
Number of labeled split graphs on n vertices such that {1..k} is independent and {k+1..n} is a clique for some k in {0..n}.
Original entry on oeis.org
1, 2, 6, 24, 128, 928, 9280, 129152, 2515200, 68780544, 2647000064, 143580989440, 10988411686912, 1187350176604160, 181232621966082048, 39089521693818912768, 11916533065969825808384, 5135497592471003032846336, 3128995097443083790244380672, 2695613904312277811648715554816
Offset: 1
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seq(1 + add((2^k-1)*2^((n-1-k)*k),k=1..n-1),n=1..20); # Georg Fischer_, May 28 2024
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def f(n): return 1+sum((2**k-1)*2**((n-1-k)*k) for k in range(1,n))
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