A047970 -id:A047970 - cp's OEIS frontend

A125108 Column sums of a Gaussian polynomial-shaped array. Row sums generate the Eulerian array A008292.

1, 2, 4, 10, 26, 72, 202, 580

Offset: 1

Alford Arnold, Dec 25 2006

Column sums of the Gaussian polynomial template count numeric partitions. Row sums of the Gaussian polynomial template generate Pascal's triangle. A105552 has the same shape as the template and counts compositions. Row sums of the Eulerian array counts permutations of n object.

			The column sums begin 1 2 4 10 26 72 202 580 ... because the structure of the Array begin as follows:
1..................................................................
......1............................................................
......1............................................................
............1......................................................
............2......2...............................................
............1......................................................
..................1................................................
..................3......5......3..................................
..................3......5......3..................................
..................1................................................
............................1......................................
............................4.......9.......9.......4..............
............................6.......16......22......16.......6.....
............................4.......9.......9.......4..............
............................1......................................
etc.

Cf. A000041, A000079, A000142, A007318, A008292, A047970, A060351, A105552.

A346661 Number of cyclic patterns of length n that avoid the vincular pattern 23-4-1.

Original entry on oeis.org

1, 1, 2, 5, 15, 50, 180, 690, 2792, 11857, 52633, 243455, 1170525, 5837934, 30151474, 161021581, 888001485, 5051014786, 29600662480, 178541105770, 1107321666920, 7055339825171, 46142654894331, 309513540865544, 2127744119042216, 14979904453920111, 107932371558460341, 795363217306369817, 5990768203554158167, 46094392105916344968, 362092868720288824992

Offset: 0

Rupert Li, Aug 03 2021

nonn

The vincular pattern 23-4-1 requires the 2 and the 3 to be adjacent.

By the trivial Wilf equivalence obtained by reversing the permutations, a(n) is also the number of cyclic patterns of length n that avoid the vincular pattern 32-1-4.

Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021.
Toufik Mansour and Mark Shattuck, Enumerating circular permutations avoiding the vincular pattern 23 4 1, arXiv:2111.04211 [math.CO], 2021.
Toufik Mansour and Mark Shattuck, On a question of Li concerning an uncounted class of circular permutations, The Australasian Journal of Combinatorics, volume 83 part 1, 2022, pp. 176-195.

Cf. A025242, A047970, A346660.

More terms from Vaclav Kotesovec, Nov 09 2021, computed by Toufik Mansour and Mark Shattuck

cp's OEIS Frontend

A125108 Column sums of a Gaussian polynomial-shaped array. Row sums generate the Eulerian array A008292.

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