A348702 Square array T(n, k) (n>=1, k>=1) read by antidiagonals upwards. T(n, k) is the number of partitions of the set [n] into lists of k noncrossing sets.
1, 1, 2, 1, 4, 3, 1, 8, 9, 4, 1, 14, 27, 16, 5, 1, 22, 75, 64, 25, 6, 1, 32, 183, 244, 125, 36, 7, 1, 44, 393, 844, 605, 216, 49, 8, 1, 58, 759, 2584, 2725, 1266, 343, 64, 9, 1, 74, 1347, 6976, 11105, 7026, 2359, 512, 81, 10, 1, 92, 2235, 16804, 40325, 35976, 15547, 4040, 729, 100, 11, 1, 112, 3513, 36724, 129925, 166956, 95977
Offset: 1
Examples
T(4, 3) = 75. There are 3 lists with set sizes 4, 0 and 0: ({1, 2, 3, 4}, {}, {}), ..., ({}, {}, {1, 2, 3, 4}). There are 4*6 lists with set sizes 3, 1 and 0: ({1, 2, 3}, {4}, {}), ..., ({}, {1}, {2, 3, 4}). There are 6 lists with set sizes 2, 2 and 0 where 1 and 2 are in the same set: ({1, 2}, {3, 4}, {}), ..., ({}, {3, 4}, {1, 2}). There are 6 lists with set sizes 2, 2 and 0 where 1 and 4 are in the same set: ({1, 4}, {2, 3}, {}), ..., ({}, {2, 3}, {1, 4}). There are 6*6 lists with set sizes 2, 1 and 1: ({1, 2}, {3}, {4}), ..., ({2}, {1}, {3, 4}). When adding the 6 list of crossing sets, lists with set sizes 2, 2 and 0 where 1 and 3 are in the same set, ({1, 3}, {2, 4}, {}), ..., ({}, {2, 4}, {1, 3}), then we have 81 partitions of {1, 2, 3, 4} into lists of sets. This is found in A089072(4, 3) = 81.
Links
- David Callan, Sets, Lists and Noncrossing Partitions, arXiv:0711.4841 [math.CO], 2007-2008.
Comments