A340106 -id:A340106 - cp's OEIS frontend

A340107 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive.

1, 1, 1, 1, 2, 2, 1, 3, 6, 0, 1, 4, 12, 16, 16, 1, 5, 20, 50, 90, 80, 1, 6, 30, 108, 300, 552, 516, 1, 7, 42, 196, 742, 2100, 3990, 3794, 1, 8, 56, 320, 1536, 5888, 16976, 32656, 31456, 1, 9, 72, 486, 2826, 13680, 53046, 154350, 299628, 290970, 1, 10, 90, 700, 4780, 27960, 136380, 532340, 1559040, 3044900, 2974380

Offset: 0

Xiangyu Chen, Dec 28 2020

In a convex n-gon, the number of paths using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.

			n\k   0     1     2     3     4     5     6     7     8
0     1
1     1     1
2     1     2      2
3     1     3      6      0
4     1     4     12     16     16
5     1     5     20     50     90    80
6     1     6     30    108    300   552    516
7     1     7     42    196    742  2100   3990  3794
8     1     8     56    320   1536  5888  16976 32656 31456

Cf. A338526, A338838, A338849.

Cf. A340106, A340108.

T(n,k) = n*(A340106(n-1,k-1) - S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0. [exception added by Xiangyu Chen, Aug 19 2022]

A340108 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] in a circle with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive, and rotations are considered to be distinct.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 0, 1, 4, 12, 0, 16, 1, 5, 20, 30, 80, 60, 1, 6, 30, 84, 264, 480, 456, 1, 7, 42, 168, 672, 1890, 3612, 3458, 1, 8, 56, 288, 1424, 5440, 15744, 30352, 29296, 1, 9, 72, 450, 2664, 12870, 50004, 145656, 283104, 275166, 1, 10, 90, 660, 4560, 26640, 130080, 508060, 1488960, 2909700, 2843980

Offset: 0

Xiangyu Chen, Dec 28 2020

In a convex n-gon, the number of cycles using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.

			n\k   0     1      2      3     4     5     6     7     8
0     1
1     1     1
2     1     2      2
3     1     3      6      0
4     1     4     12      0     16
5     1     5     20     30     80    60
6     1     6     30     84    264   480    456
7     1     7     42    168    672  1890   3612  3458
8     1     8     56    288   1424  5440  15744 30352 29296

Cf. A338526, A338838, A338849.

Cf. A340106, A340107.

T(n,k) = n*(5*A340106(n-1,k-1) - 2*Z(n,k) - Z(n-1,k-1) - 2*S(n,k) - 2*S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0, Z(n,k) = 2*A340106(n-1,k-1) - S(n,k) - V(n-1,k-1), Z(n,k)=0 for k <= 0, V(n,k) = Z(n-1,k-1) - V(n-1,k-1), V(n,k)=0 for k <= 0 except for V(2,2)=2.

Terms of column 2 corrected by Xiangyu Chen, Aug 19 2022

cp's OEIS Frontend

A340107 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive.

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