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Triangle T(n,k) of number of strongly connected digraphs on n unlabeled nodes and with k arcs, k=0..n*(n-1).
+10
10
1, 0, 0, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 4, 16, 22, 22, 11, 5, 1, 1, 0, 0, 0, 0, 0, 1, 7, 58, 240, 565, 928, 1065, 953, 640, 359, 150, 59, 16, 5, 1, 1, 0, 0, 0, 0, 0, 0, 1, 10, 165, 1281, 6063, 19591, 47049, 87690, 131927, 163632, 170720, 151238, 115122, 75357, 42745, 20891, 8877, 3224, 1039, 286, 76, 17, 5, 1, 1
EXAMPLE
Triangle begins:
[1] 1;
[2] 0,0,1;
[3] 0,0,0,1,2,1,1;
[4] 0,0,0,0,1,4,16,22,22,11,5,1,1;
...
The number of strongly connected digraphs on 3 unlabeled nodes is 5 = 1+2+1+1.
PROG
(PARI) \\ See PARI link in A350489 for program code. { my(A=A057276rows(5)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 13 2022
Triangle T(n,k) of number of digraphs with a source and a sink on n unlabeled nodes and k arcs, k=0..n*(n-1).
+10
9
1, 0, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 1, 11, 31, 45, 38, 27, 13, 5, 1, 1, 0, 0, 0, 0, 1, 23, 152, 486, 992, 1419, 1641, 1485, 1152, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 0, 1, 42, 517, 3194, 12174, 32860, 68423, 116168, 166164, 204867, 219906, 206993, 170922, 124088, 78809, 43860, 21209, 8951, 3242, 1043, 288, 76, 17, 5, 1, 1
REFERENCES
V. Jovovic, G. Kilibarda, Enumeration of labeled initially-finally connected digraphs, Scientific review, Serbian Scientific Society, 19-20 (1996), p. 246.
EXAMPLE
Triangle begins:
[1],
[0,1,1],
[0,0,1,4,4,1,1],
[0,0,0,1,11,31,45,38,27,13,5,1,1],
...
The number of digraphs with a source and a sink on 3 unlabeled nodes is 11 = 1+4+4+1+1.
PROG
(PARI) \\ See PARI link in A350794 for program code. { my(A=A057278triang(5)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 21 2022
Triangle T(n,k) of number of digraphs with a source on n unlabeled nodes with k arcs, k=0..n*(n-1).
+10
8
1, 0, 1, 1, 0, 0, 2, 4, 4, 1, 1, 0, 0, 0, 4, 16, 34, 46, 38, 27, 13, 5, 1, 1, 0, 0, 0, 0, 9, 56, 229, 573, 1058, 1448, 1653, 1487, 1153, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 0, 20, 198, 1218, 5089, 15596, 37302, 72776, 119531, 168233, 205923, 220337, 207147, 170965, 124099, 78811, 43861, 21209, 8951, 3242, 1043, 288, 76, 17, 5, 1, 1
EXAMPLE
Triangle begins:
[1],
[0, 1, 1],
[0, 0, 2, 4, 4, 1, 1],
[0, 0, 0, 4, 16, 34, 46, 38, 27, 13, 5, 1, 1],
....
The number of digraphs with a source on 3 unlabeled nodes is 12 = 2+4+4+1+1.
PROG
(PARI) \\ See PARI link in A350794 for program code. { my(A=A057277triang(5)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 21 2022
Triangle T(n,k) of number of unilaterally connected digraphs on n unlabeled nodes with k arcs, k=0..n*(n-1).
+10
7
1, 0, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 1, 10, 30, 45, 38, 27, 13, 5, 1, 1, 0, 0, 0, 0, 1, 20, 136, 462, 972, 1412, 1639, 1485, 1152, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 0, 1, 35, 437, 2833, 11325, 31615, 67207, 115344, 165762, 204723, 219866, 206986, 170920, 124088, 78809, 43860, 21209, 8951, 3242, 1043, 288, 76, 17, 5, 1, 1
REFERENCES
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.
EXAMPLE
[1],[0,1,1],[0,0,1,4,4,1,1],[0,0,0,1,10,30,45,38,27,13,5,1,1],...; Number of unilaterally connected digraphs on 4 unlabeled nodes is 171=1+10+30+45+38+27+13+5+1+1.
Triangle T(n,k) of number of digraphs with a quasi-source on n labeled nodes and with k arcs, k=0,1,..,n*(n-1).
+10
5
1, 0, 2, 1, 0, 0, 12, 20, 15, 6, 1, 0, 0, 0, 104, 426, 768, 920, 792, 495, 220, 66, 12, 1, 0, 0, 0, 0, 1160, 9184, 32420, 73000, 123425, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1
EXAMPLE
Triangle starts:
1;
0,2,1;
0,0,12,20,15,6,1;
0,0,0,104,426,768,920,792,495,220,66,12,1;
...
Number of digraphs with a quasi-source on 3 labeled nodes is 54=12+20+15+6+1.
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