A086852 -id:A086852 - cp's OEIS frontend

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.

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.

A384494 Triangle read by rows: T(n, k) = (-1)^k(k+1)(n+1-k)!, n >= 0, k = 0..n.

Original entry on oeis.org

1, 2, -2, 6, -4, 3, 24, -12, 6, -4, 120, -48, 18, -8, 5, 720, -240, 72, -24, 10, -6, 5040, -1440, 360, -96, 30, -12, 7, 40320, -10080, 2160, -480, 120, -36, 14, -8, 362880, -80640, 15120, -2880, 600, -144, 42, -16, 9, 3628800, -725760, 120960, -20160, 3600, -720, 168, -48, 18, -10

Offset: 0

Wolfdieter Lang, May 31 2025

This triangle, written as (infinite) square matrix MT with vanishing upper diagonals 0, together with the Riordan triangle A104698, written also as such a square matrix MR, appears in the double sum formula for the number of certain restricted permutations given in A086852(n), as diagonal sequence A086852(n+2) = (2*MR*MT^t)_{n,n}, for n >=0, where t indicates matrix transpositon.

			The triangle T begins:
  n\k        0        1       2       3     4     5   6   7  8    9 ...
  ---------------------------------------------------------------------
  0:         1
  1:         2       -2
  2:         6       -4       3
  3:        24      -12       6      -4
  4:       120      -48      18      -8     5
  5:       720     -240      72     -24    10    -6
  6:      5040    -1440     360     -96    30   -12   7
  7:     40320   -10080    2160    -480    12   -36  14  -8
  8:    362880   -80640   15120   -2880   600  -144  42 -16  9
  9:   3628800  -725760  120960  -20160  3600  -720 168 -48 18 -10
  ...

Cf. A104698, A086852, A086852.

Column sequences: A000142(n+1), -A052849, A052560(n-1), -A052578(n-2), A052648(n-3), -A298881(n-4), A062098(n-5), -A159038(n-6), ...

Table[(-1)^k * (k+1) * (n+1-k)!, {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 31 2025 *)

T(n, k) = (-1)^k*(k+1)*(n+1-k)!, for n >= 0 and k = 0, 1, ..., n.

O.g.f. of row polynomials P(n, y) := Sum_{k=0..n} T(n, k) y^k: G(x, y) = ((N(x) - 1)/x) * (1/(1 + y*x)^2), with N(x) = hypergeometric([1,1], [], x), the o.g.f. of {n!}_{n>=0} (see A000142).

A384186 Number of permutations of 1, 2,..., n with exactly one rising or falling successon, namely (n-1)n or n(n-1).

Original entry on oeis.org

0, 2, 2, 2, 6, 34, 214, 1506, 11990, 107234, 1065846, 11659426, 139217494, 1801784610, 25124797046, 375531165794, 5989287277014, 101524201538146, 1822662037112950, 34548339122512674, 689469487015534166, 14450128299126915746

Offset: 1

Wolfdieter Lang, May 21 2025

For the number of permutations of length n with exactly one rising or falling successon see A086852. For the number of such permutations without either (n-1)n or n(n-1) see A383857, for n >= 1.

			a(2) = 2*1 from 12 and the reverted 21.
a(3) = 2*1 from 132 and 231.
a(4) = 2*1 from 1342 and 2431.
a(5) = 2*3 from 24513, 24531, 31452 and 31542, 13542, 25413.

Cf. A002464, A086652, A383857.

a(n) = A086652(n) - A383857(n), for n >= 1.
a(n) = a(n-2) + 2*(n-2)*A002464(n-2) + 2*A383857(n-2), for n >= 3, with a(1) = 0 and a(2) = 2. One could also use this recurrence for n >= 2, using a(0) = -2 and a(1) = 0.
a(n) = a(n-2) + 2*(b(n-1) + b(n-2)), with b = A002464, for n >= 3, with a(1) = 0 and a(2) = 2.

cp's OEIS Frontend

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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A384494 Triangle read by rows: T(n, k) = (-1)^k(k+1)(n+1-k)!, n >= 0, k = 0..n.

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A384186 Number of permutations of 1, 2,..., n with exactly one rising or falling successon, namely (n-1)n or n(n-1).

Views

Author

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Formula

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A384494 Triangle read by rows: T(n, k) = (-1)^k*(k+1)*(n+1-k)!, n >= 0, k = 0..n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A384186 Number of permutations of 1, 2,..., n with exactly one rising or falling successon, namely (n-1)n or n(n-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A384494 Triangle read by rows: T(n, k) = (-1)^k(k+1)(n+1-k)!, n >= 0, k = 0..n.