A363181 -id:A363181 - cp's OEIS frontend

A211694 Number of partitions of [n] that contain no isolated singletons.

1, 0, 1, 1, 2, 3, 6, 11, 23, 47, 103, 226, 518, 1200, 2867, 6946, 17234, 43393, 111419, 290242, 768901, 2065172, 5630083, 15549403, 43527487, 123343911, 353864422, 1026935904, 3014535166, 8945274505, 26829206798, 81293234754, 248805520401, 768882019073, 2398686176048, 7552071250781

Offset: 0

R. H. Hardin, Apr 19 2012

nonn

Number of nonnegative integer arrays of length n with new values introduced in order 0 upwards and every value appearing only in runs of at least 2.

Column 2 of A211700.

			All solutions for n = 7:
  0    0    0    0    0    0    0    0    0    0    0
  0    0    0    0    0    0    0    0    0    0    0
  0    0    1    0    1    0    0    0    1    1    1
  1    1    1    1    1    0    0    0    1    1    1
  1    1    2    1    1    0    1    0    0    1    1
  2    1    2    0    2    0    1    1    0    1    0
  2    1    2    0    2    0    1    1    0    1    0

Alois P. Heinz, Table of n, a(n) for n = 0..1132
A. O. Munagi, Set partitions with isolated singletons, Am. Math. Monthly 125 (2018), 447-452.

Cf. A160823, A211700.

Cf. A198648, A198655, A303587, A303588, A363181.

f:=proc(n) local j;
add(combinat:-bell(j-1)*binomial(n-j-1, j-1), j=0..floor(n/2));
end;
[seq(f(n), n=0..100)]; # N. J. A. Sloane, May 19 2018

a[n_] := If[n == 0, 1, Sum[BellB[j-1]*Binomial[n-j-1, j-1], {j, 1, Floor[n/2]}]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 17 2024, after Maple code *)

G.f.: 1+x^2/W(0), where W(k) = 1 - x - x^2/(1 - x^2*(k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2014

Edited by Andrey Zabolotskiy, Feb 07 2025

A229730 Number of separable permutations with the maximum number of occurrences of the 1-box pattern on separable permutations.

Original entry on oeis.org

0, 0, 2, 2, 8, 14, 54, 128, 466

Offset: 0

N. J. A. Sloane, Oct 01 2013

See Kitaev-Remmel for precise definition.

S. Kitaev and J. Remmel, The 1-box pattern on pattern avoiding permutations, arXiv preprint arXiv:1305.6970 [math.CO], 2013.

Cf. A229729, A363181.

A363236 Number of permutations p of [n] such that each element in p has at least one neighbor with opposite parity.

Original entry on oeis.org

1, 0, 2, 2, 16, 36, 288, 1152, 10368, 57600, 604800, 4320000, 51840000, 453600000, 6147187200, 63605606400, 962415820800, 11500218777600, 192255565824000, 2605984690176000, 47721518530560000, 723526168780800000, 14407079038894080000, 241602987041095680000

Offset: 0

Alois P. Heinz, May 22 2023

nonn

			a(0) = 1: (), the empty permutation.
a(1) = 0.
a(2) = 2: 12, 21.
a(3) = 2: 123, 321.
a(4) = 16: 1234, 1243, 1423, 1432, 2134, 2143, 2314, 2341, 3214, 3241, 3412, 3421, 4123, 4132, 4312, 4321.
a(5) = 36: 12345, 12354, 12534, 12543, 14325, 14352, 14523, 14532, 21345, 21543, 23145, 23541, 25143, 25341, 32145, 32154, 32514, 32541, 34125, 34152, 34512, 34521, 41325, 41523, 43125, 43521, 45123, 45321, 52134, 52143, 52314, 52341, 54123, 54132, 54312, 54321.

Alois P. Heinz, Table of n, a(n) for n = 0..465
Wikipedia, Permutation

Cf. A001622, A092186, A363180, A363181.

a(n) ~ phi^n * n! / (5^(1/4) * 2^(n-1)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 26 2023

A346462 Triangle read by rows: T(n,k) gives the number of permutations of length n containing exactly k instances of the 1-box pattern; 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 0, 0, 4, 2, 2, 0, 10, 4, 8, 14, 0, 40, 10, 42, 14, 90, 0, 230, 40, 226, 80, 54, 646, 0, 1580, 230, 1480, 442, 534, 128, 5242, 0, 12434, 1580, 11496, 2920, 4746, 1404, 498, 47622, 0, 110320, 12434, 101966, 22762, 45216, 13138, 7996, 1426

Offset: 0

Peter Kagey, Jul 19 2021

An instance of the 1-box pattern in a permutation pi is a letter pi_i such that pi_{i-1} or pi_{i+1} differs from pi_i by exactly 1.
Column k=0 is A002464. Columns k=2 and k=3 are given by A086852.
Main diagonal begins: 1,0,2,2,8,14,54,128,498,1426,5736,... A363181.

			The permutation 14327568 has 5 instances of the 1-box pattern:
- position 2 differs from position 3 by one,
- position 3 differs from positions 2 and 4 by one,
- position 4 differs from position 3 by one,
- position 6 differs from position 7 by one,
- position 7 differs from position 6 by one, and
positions 1, 5, and 8 differ from all of their neighbors by more than 1.
Table begins:
  n\k|  0  1    2   3    4   5   6
-----+-----------------------------
   0 |  1
   1 |  1  0
   2 |  0  0    2
   3 |  0  0    4   2
   4 |  2  0   10   4    8
   5 | 14  0   40  10   42  14
   6 | 90  0  230  40  226  80  54

Alois P. Heinz, Rows n = 0..20, flattened
FindStat, St000064: The number of one-box pattern of a permutation.
S. Kitaev, J. Remmel, The 1-box pattern on pattern avoiding permutations, arXiv:1305.6970 [math.CO], 2013.

Cf. A002464, A086852, A229729, A229730, A363181.

Row sums give A000142.

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A211694 Number of partitions of [n] that contain no isolated singletons.

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A229730 Number of separable permutations with the maximum number of occurrences of the 1-box pattern on separable permutations.

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A363236 Number of permutations p of [n] such that each element in p has at least one neighbor with opposite parity.

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A346462 Triangle read by rows: T(n,k) gives the number of permutations of length n containing exactly k instances of the 1-box pattern; 0 <= k <= n.

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