A046858 Irregular triangle read by rows: T(n,k) = number of directed graphs-with-loops with n nodes and k arcs (n >= 0, 0 <= k <= n^2).
1, 1, 1, 1, 2, 4, 2, 1, 1, 2, 8, 17, 24, 24, 17, 8, 2, 1, 1, 2, 9, 32, 95, 203, 373, 515, 584, 515, 373, 203, 95, 32, 9, 2, 1, 1, 2, 9, 36, 157, 549, 1692, 4374, 9626, 17874, 28373, 38486, 44805, 44805, 38486, 28373, 17874, 9626, 4374, 1692, 549, 157, 36, 9, 2, 1, 1
Offset: 0
Examples
Triangle begins: [1], [1, 1], [1, 2, 4, 2, 1], [1, 2, 8, 17, 24, 24, 17, 8, 2, 1], [1, 2, 9, 32, 95, 203, 373, 515, 584, 515, 373, 203, 95, 32, 9, 2, 1] (the last batch giving the numbers of directed graphs with loops on 4 nodes and from 0 to 16 arcs).
References
- W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
Links
- W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
Crossrefs
Cf. A000595.
Programs
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Mathematica
Needs["Combinatorica`"]; Join[{{1}, {1,1}}, CoefficientList[Table[CycleIndex[Join[PairGroup[SymmetricGroup[n], Ordered], Permutations[Range[n^2-n+1, n^2]], 2],s]/.Table[s[i]->1+x^i, {i,1,n^2-n}], {n,2,7}], x]]//Grid (* Geoffrey Critzer, Sep 29 2012 *)
Extensions
More terms from Vladeta Jovovic, Feb 07 2000
Edited by N. J. A. Sloane Apr 16 2008 at the suggestion of Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 24 2008
Comments