Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.
All indices begin at 0. Sequences with ellipses have additional terms in the OEIS. Terms I disagree with are followed by my preferred alternative in square brackets.
*** Clutters ***
A clutter is a connected antichain of finite nonempty sets (edges). It is spanning if all n vertices are covered by some edge. A clutter with singleton edges allowed is one whose non-singleton edges only are required to form an antichain.
Labeled:
A305005 (not spanning without singletons): 1, 1, 2, 9, 111, 6829, 7783192, ...
A048143 (spanning without singletons): 1, 1[0], 1, 5, 84, 6348, 7743728, ...
A304984 (not spanning with singletons): 1, 2, 7, 56, 1533, 210302, 496838435, ...
A304985 (spanning with singletons): 1, 1, 4, 40, 1344, 203136, 495598592, ...
Unlabeled:
A304981 (not spanning without singletons): 1, 1, 2, 5, 19, 176, 16118, 489996568
A261006 (spanning without singletons): 1, 1[0], 1, 3, 14, 157, 15942, 489980450
A304982 (not spanning with singletons): 1, 2, 5, 19, 137
A304983 (spanning with singletons): 1, 1, 3, 14, 118
*** Antichains ***
An antichain is a finite set of finite nonempty sets (edges), none of which is a subset of any other. It is spanning if all n vertices are covered by some edge. An antichain with singleton edges allowed is one whose non-singleton edges only are required to form an antichain.
Labeled:
A006126 (not spanning without singletons): 1, 1, 2, 9, 114, 6894, 7785062, ...
A305001 (spanning without singletons): 1, 0, 1, 5, 87, 6398, 7745253, ...
A305000 (not spanning with singletons): 1, 2, 8, 72, 1824
A304999 (spanning with singletons): 1, 1, 5, 53, 1577
Unlabeled:
A261005 (not spanning without singletons): 1, 1, 2, 5, 20, 180, 16143, 489996795
A304998 (spanning without singletons): 1, 0, 1, 3, 15, 160, 15963, 489980652
A304996 (not spanning with singletons): 1, 2, 6, 24, 166
A304997 (spanning with singletons): 1, 1, 4, 18, 142
*** Hypertrees ***
A hypertree is a connected antichain of finite nonempty sets (branches) with no cycles, or equivalently, whose clutter density is -1. It is spanning if all n vertices are covered by some branch. A hypertree with singleton branches allowed is one whose non-singleton branches only are required to form an antichain.
Labeled:
A305004 (not spanning without singletons): 1, 1, 2, 8, 52, 507, 6844, 118582, ...
A030019 (spanning without singletons): 1, 1[0], 1, 4, 29, 311, 4447, 79745, ...
A304968 (not spanning with singletons): 1, 2, 7, 48, 621, 12638, 351987, ...
A134958 (spanning with singletons): 1, 2[1], 4, 32, 464, 9952, 284608, ...
Unlabeled:
A304970 (not spanning without singletons): 1, 1, 2, 4, 8, 17, 39, 98
A035053 (spanning without singletons): 1, 1[0], 1, 2, 4, 9, 22, 59, ...
A304386 (not spanning with singletons): 1, 2, 5, 15, 50, 200, 907
A134959 (spanning with singletons): 1, 1, 3, 10, 35, 150, 707
*** Hyperforests ***
A hyperforest is an antichain of finite nonempty sets (branches) whose connected components are hypertrees. It is spanning if all n vertices are covered by some branch. A hyperforest with singleton branches allowed is one whose non-singleton branches only are required to form an antichain.
Labeled:
A134954 (not spanning without singletons): 1, 1, 2, 8, 55, 562, 7739, 134808, ...
A304911 (spanning without singletons): 1, 0, 1, 4, 32, 351, 5057, 90756, ...
A134956 (not spanning with singletons): 1, 2, 8, 64, 880, 17984, 495296, ...
A304919 (spanning with singletons): 1, 1, 5, 45, 665, 14153, 399421, ...
Unlabeled:
A134955 (not spanning without singletons): 1, 1, 2, 4, 9, 20, 50, 128, 351, ...
A144959 (spanning without singletons): 1, 0, 1, 2, 5, 11, 30, 78, 223, ...
A134957 (not spanning with singletons): 1, 2, 6, 20, 75, 310, 1422
A304977 (spanning with singletons): 1, 1, 4, 14, 55, 235, 1112