A000313 -id:A000313 - cp's OEIS frontend

A010027 Triangle read by rows: T(n,k) is the number of permutations of [n] having k consecutive ascending pairs (0 <= k <= n-1).

1, 1, 1, 1, 2, 3, 1, 3, 9, 11, 1, 4, 18, 44, 53, 1, 5, 30, 110, 265, 309, 1, 6, 45, 220, 795, 1854, 2119, 1, 7, 63, 385, 1855, 6489, 14833, 16687, 1, 8, 84, 616, 3710, 17304, 59332, 133496, 148329, 1, 9, 108, 924, 6678, 38934, 177996, 600732, 1334961, 1468457, 1

Offset: 1

N. J. A. Sloane

A "consecutive ascending pair" in a permutation p_1, p_2, ..., p_n is a pair p_i, p_{i+1} = p_i + 1.
From Emeric Deutsch, May 15 2010: (Start)
The same triangle, but with rows indexed differently, also arises as follows: U(n,k) = number of permutations of [n] having k blocks (1 <= k <= n), where a block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67.
When seen as coefficients of polynomials with decreasing exponents: evaluations are A001339 (x=2), A081923 (x=3), A081924 (x=4), A087981 (x=-1).
The sum of the entries in row n is n!.
U(n,n) = A000255(n-1) = d(n-1) + d(n), U(n,n-1)=d(n), where d(j)=A000166(j) (derangement numbers). (End)
This is essentially the reversal of the exponential Riordan array [exp(-x)/(1-x)^2,x] (cf. A123513). - Paul Barry, Jun 17 2010
U(n-1, k-2) * n*(n-1)/k = number of permutations of [n] with k elements not fixed by the permutation. - Michael Somos, Aug 19 2018

			Triangle starts:
  1;
  1, 1;
  1, 2,   3;
  1, 3,   9,  11;
  1, 4,  18,  44,   53;
  1, 5,  30, 110,  265,   309;
  1, 6,  45, 220,  795,  1854,   2119;
  1, 7,  63, 385, 1855,  6489,  14833,  16687;
  1, 8,  84, 616, 3710, 17304,  59332, 133496,  148329;
  1, 9, 108, 924, 6678, 38934, 177996, 600732, 1334961, 1468457;
  ...
For n=3, the permutations 123, 132, 213, 231, 312, 321 have respectively 2,0,0,1,1,0 consecutive ascending pairs, so row 3 of the triangle is 3,2,1. - _N. J. A. Sloane_, Apr 12 2014
In the alternative definition, T(4,2)=3 because we have 234.1, 4.123, and 34.12 (the blocks are separated by dots). - _Emeric Deutsch_, May 16 2010

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Alois P. Heinz, Rows n = 1..150, flattened
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357. [_Emeric Deutsch_, May 15 2010]

Diagonals, reading from the right-hand edge: A000255, A000166, A000274, A000313, A001260, A001261. A045943 is another diagonal.
Cf. A123513 (mirror image).
A289632 is the analogous triangle with the additional restriction that all consecutive pairs must be isolated, i.e., must not be back-to-back to form longer consecutive sequences.

U := proc (n, k) options operator, arrow: factorial(k+1)*binomial(n, k)*(sum((-1)^i/factorial(i), i = 0 .. k+1))/n end proc: for n to 10 do seq(U(n, k), k = 1 .. n) end do; # yields sequence in triangular form; # Emeric Deutsch, May 15 2010

t[n_, k_] := Binomial[n, k]*Subfactorial[k+1]/n; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 07 2014, after Emeric Deutsch *)
T[0,0]:=0; T[1,1]:=1; T[n_,n_]:=T[n,n]=(n-1)T[n-1,n-1]+(n-2)T[n-2,n-2]; T[n_,k_]:=T[n,k]=T[n-1,k] (n-1)/(n-k); Flatten@Table[T[n,k],{n,1,10},{k,1,n}] (* Oliver Seipel, Dec 01 2024 *)

E.g.f.: exp(x*(y-1))/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
From Emeric Deutsch, May 15 2010: (Start)
U(n,k) = binomial(n-1,k-1)*(k-1)!*Sum_{j=0..k-1} (-1)^(k-j-1)*(j+1)/(k-j-1)! (1 <= k <= n).
U(n,k) = (k+1)!*binomial(n,k)*(1/n)*Sum_{i=0..k+1} (-1)^i/i!.
U(n,k) = (1/n)*binomial(n,k)*d(k+1), where d(j)=A000166(j) (derangement numbers). (End)

More terms from Vladeta Jovovic, Jan 03 2003

Original definition from David, Kendall and Barton restored by N. J. A. Sloane, Apr 12 2014

A123513 Triangle read by rows: T(n,k) is the number of permutations of [n] having k small descents (n >= 1; 0 <= k <= n-1). A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) = 1.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 11, 9, 3, 1, 53, 44, 18, 4, 1, 309, 265, 110, 30, 5, 1, 2119, 1854, 795, 220, 45, 6, 1, 16687, 14833, 6489, 1855, 385, 63, 7, 1, 148329, 133496, 59332, 17304, 3710, 616, 84, 8, 1, 1468457, 1334961, 600732, 177996, 38934, 6678, 924, 108, 9, 1

Offset: 1

Emeric Deutsch, Oct 02 2006

This triangle is essentially A010027 (ascending pairs in permutations of [n]) with a different offset. The same triangle gives the number of permutations of [n] having k unit ascents (n >= 1; 0 <= k <= n-1). For permutations sorted by number of non-unitary (i.e., > 1) descents (also called "big" descents), see A120434. For permutations sorted by number of unitary moves (i.e., ascent or descent), see A001100. - Olivier Gérard, Oct 09 2007

With offsets n=0 (k=0) this is a binomial convolution triangle, a Sheffer triangle of the Appell type: ((exp(-x))/(1-x)^2),x). See the e.g.f. given below.

			Triangle starts:
     1;
     1,    1;
     3,    2,   1;
    11,    9,   3,   1;
    53,   44,  18,   4,  1;
   309,  265, 110,  30,  5, 1;
  2119, 1854, 795, 220, 45, 6, 1;
  ...
T(4,2)=3 because we have 14/3/2, 2/14/3 and 3/2/14 (the unit descents are shown by a /).
T(4,2)=3 because we have 14/3/2, 2/14/3 and 3/2/14 (the small descents are shown by a /).

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 179, Table 5.4 for S_{n,k} (without row n=1 and column k=0).
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263 (Table 7.5.1).

Alois P. Heinz, Rows n = 1..150, flattened
Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016. See Table 0.
FindStat - Combinatorial Statistic Finder, The number of adjacencies of a permutation
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
J. Liese and J. Remmel, Q-analogues of the number of permutations with k-excedances, PU. M. A. Vol. 21 (2010), No. 2, pp. 285-320 (see E_{n,1}(x) in Table 1 p. 291).
F. Poussin, Énumération des permutations par nombre de marches, RAIRO, Informatique théorique, 13 no. 3, 1979, p. 251-255.

Cf. A000166, A000255, A000274, A000313, A001260.

Cf. A010027 (mirror image), A120434, A001100.

G:=exp(-x+t*x)/(1-x)^2: Gser:=simplify(series(G,x=0,15)): for n from 0 to 10 do P[n+1]:=sort(n!*coeff(Gser,x,n)) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=0..n-1) od; # yields sequence in triangular form

Needs["Combinatorica`"];
Table[Map[Count[#,1]&,Map[Differences,Permutations[n]]]//Distribution,{n,1,10}]//Grid
(* Geoffrey Critzer, Dec 15 2012 *)
T[n_, k_] := (n-1)! SeriesCoefficient[Exp[-x + t x]/(1-x)^2, {x, 0, n-1}, {t, 0, k}];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 25 2019 *)
T[1,1]:=1;T[0,1]:=0;T[n_,1]:=T[n,1]=(n-1)T[n-1,1]+(n-2)T[n-2,1];T[n_,k_]:=T[n,k]=T[n-1, k-1](n-1)/(k-1);Flatten@Table[T[n,k],{n,1,10},{k,1,n}] (* Oliver Seipel, Dec 01 2024 *)

T(n,1) = A000255(n-1).
T(n,2) = A000166(n-1) (the derangement numbers).
T(n,3) = A000274(n).
T(n,4) = A000313(n).
T(n,5) = A001260(n);
G.f.: exp(-x+tx)/(1-x)^2 (if offset is 0), i.e., T(n,k)=(n-1)!*[x^(n-1) t^k]exp(-x+tx)/(1-x)^2.
T(n,k) = binomial(n-1,k)*A000255(n-1), n-1 >= k >= 0, else 0.

A000274 Number of permutations of length n with 2 consecutive ascending pairs.

Original entry on oeis.org

0, 0, 1, 3, 18, 110, 795, 6489, 59332, 600732, 6674805, 80765135, 1057289046, 14890154058, 224497707343, 3607998868005, 61576514013960, 1112225784377144, 21197714949305577, 425131949816628507, 8950146311929021210, 197350726178034917670, 4548464355722328578691

Offset: 1

N. J. A. Sloane, Simon Plouffe

From Emeric Deutsch, May 25 2009: (Start)
a(n) = number of excedances in all derangements of [n-1]. Example: a(5)=18 because the derangements of {1,2,3,4} are 4*123, 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 2*3*4*1, 3*4*21, 4*3*21 with the 18 excedances marked. An excedance of a permutation p is a position i such that p(i)>i.
a(n) = Sum(k*A046739(n,k), k>=1).
(End)
Appears to be the inverse binomial transform of A001286 (filling the two leading zeros in there), then shifting one place to the right. - R. J. Mathar, Apr 04 2012

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210 (divided by 2).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..150
R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [_Emeric Deutsch_, May 25 2009]

Cf. A010027, A000255, A000166, A000313, A001260, A001261.
A diagonal in triangle A010027.
Cf. A046739. [Emeric Deutsch, May 25 2009]
Cf. A145887, A145886.

a:= n->sum((n-1)!*sum((-1)^k/k!/2, j=1..n-1), k=0..n-1): seq(a(n), n=1..23); # Zerinvary Lajos, May 17 2007

Table[Subfactorial[n]*n/2, {n, 2, 20}] (* Zerinvary Lajos, Jul 09 2009 *)

a(n) = (1 + n) a(n - 1) + (3 + n) a(n - 2) + (3 - n) a(n - 3) + (2 - n) a(n - 4).
E.g.f.: x^2/2*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
a(n) = (n-1)^2/(n-2)*a(n-1)-(-1)^n*(n-1)/2, n>2, a(2)=0. - Vladeta Jovovic, Aug 31 2003
a(n) = (1/2){[n!/e] - [(n-1)!/e]} (conjectured).
a(n) = (n-1)*GAMMA(n,-1)*exp(-1)/2 where GAMMA = incomplete Gamma function. [Mark van Hoeij, Nov 11 2009]
a(n) = A145887(n-1) + A145886(n-1). - Anton Zakharov,  Aug 28 2016

Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014

A001260 Number of permutations of length n with 4 consecutive ascending pairs.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 45, 385, 3710, 38934, 444990, 5506710, 73422855, 1049946755, 16035550531, 260577696015, 4489954146860, 81781307674780, 1570201107355980, 31698434854748604, 671260973394676605, 14879618243581997745

Offset: 1

N. J. A. Sloane

nonn

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A010027, A000255, A000166, A000274, A000313, A001261.

A diagonal in triangle A010027.

a:=n->sum((n+2)!*sum((-1)^k/k!/4!, j=1..n), k=0..n): seq(a(n), n=2..19); # Zerinvary Lajos, May 25 2007
series(hypergeom([2, 5],[],x/(x+1))/(x+1)^5,x=0,30); # Mark van Hoeij, Nov 07 2011

Drop[CoefficientList[Series[x^4/4! Exp[-x]/(1 - x)^2, {x, 0, 20}], x] Range[0, 20]!, 4] (* Vaclav Kotesovec, Mar 26 2014 *)

(n-1)*a(n) = (n+3)*(a(n-1)*n + a(n-2)*n - a(n-1) + 2*a(n-2)).
E.g.f.: (for offset 4): (x^4/4!)*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
G.f.: (for offset 0): hypergeom([2, 5],[],x/(x+1))/(x+1)^5. - Mark van Hoeij, Nov 07 2011
Recurrence (for offset 5): (n-5)*a(n) = (n-5)*(n-1)*a(n-1) + (n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Mar 26 2014
a(n) ~ n! * exp(-1)/24. - Vaclav Kotesovec, Mar 26 2014

More terms from Vladeta Jovovic, Jan 03 2003

Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014

A001261 Number of permutations of length n with 5 consecutive ascending pairs.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 63, 616, 6678, 77868, 978978, 13216104, 190899423, 2939850914, 48106651593, 833848627248, 15265844099324, 294412707629208, 5966764207952724, 126793739418994416, 2819296088257641741, 65470320271760790078

Offset: 1

N. J. A. Sloane

nonn

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A010027, A000255, A000166, A000274, A000313, A001260.

A diagonal in triangle A010027.

a:=n->sum((n+3)!*sum((-1)^k/k!/5!, j=1..n), k=0..n): seq(a(n), n=2..19); # Zerinvary Lajos, May 25 2007

Range[0, 30]! CoefficientList[Series[x^5/5!*Exp[-x]/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)

E.g.f.: (x^5/5!)*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003

More terms from Vladeta Jovovic, Jan 03 2003

Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014

A065087 a(n) = A000166(n)*binomial(n+1,2).

Original entry on oeis.org

0, 0, 3, 12, 90, 660, 5565, 51912, 533988, 6007320, 73422855, 969181620, 13744757598, 208462156812, 3367465610145, 57727981888080, 1046800738237320, 20020064118788592, 402756584036805963, 8502638996332570140, 187953072550509445410, 4341715975916768188740

Offset: 0

N. J. A. Sloane, Nov 10 2001

nonn

a(n) is also the number of permutations of [2n-1] having n-1 isolated fixed points (i.e. adjacent entries are not fixed points). Example: a(2)=3 because we have 132, 213, and 321. - Emeric Deutsch, Apr 18 2009

Seiichi Manyama, Table of n, a(n) for n = 0..400

Equals 3 * A000313(n+2).

Cf. A000166, A000240, A000387, A305730.

a[n_] := Subfactorial[n]*Binomial[n + 1, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 18 2024 *)

a(n) = (n/2)*A000240(n+1). - Zerinvary Lajos, Dec 18 2007, corrected Jul 09 2012
a(n) = n * (n+1) * (a(n-1)/(n-1) + (-1)^n/2) for n > 1 - Seiichi Manyama, Jun 24 2018
E.g.f.: exp(-x)*x^2*(3 - 2*x + x^2)/(2*(1 - x)^3). - Ilya Gutkovskiy, Jun 25 2018

A305730 a(n) is the total displacement of all letters in all permutations of n letters with no fixed points.

Original entry on oeis.org

0, 0, 2, 8, 60, 440, 3710, 34608, 355992, 4004880, 48948570, 646121080, 9163171732, 138974771208, 2244977073430, 38485321258720, 697867158824880, 13346709412525728, 268504389357870642, 5668425997555046760, 125302048367006296940, 2894477317277845459160

Offset: 0

Seiichi Manyama, Jun 22 2018

nonn

			n | 1 2 3 4 | the displacement of all letters | a(n)
--+---------+---------------------------------+------
2 | 2 1     | 1 + 1 = 2                       |   2
3 | 2 3 1   | 1 + 1 + 2 = 4                   |   8
  | 3 1 2   | 2 + 1 + 1 = 4                   |
4 | 2 1 4 3 | 1 + 1 + 1 + 1 = 4               |  60
  | 2 3 4 1 | 1 + 1 + 1 + 3 = 6               |
  | 2 4 1 3 | 1 + 2 + 2 + 1 = 6               |
  | 3 1 4 2 | 2 + 1 + 1 + 2 = 6               |
  | 3 4 1 2 | 2 + 2 + 2 + 2 = 8               |
  | 3 4 2 1 | 2 + 2 + 1 + 3 = 8               |
  | 4 1 2 3 | 3 + 1 + 1 + 1 = 6               |
  | 4 3 1 2 | 3 + 1 + 2 + 2 = 8               |
  | 4 3 2 1 | 3 + 1 + 1 + 3 = 8               |

Seiichi Manyama, Table of n, a(n) for n = 0..400

Cf. A000166, A000313, A065087, A090672.

{a(n) = n*(n+1)!/3*sum(k=0, n, (-1)^k/k!)}

a(n) = n * (n+1) * A000166(n)/3 = 2/3 * A065087(n).
a(n) = n * (n+1)!/3 * Sum_{k=0..n} (-1)^k/k!.
a(n) = n * (n+1) * (a(n-1)/(n-1) + (-1)^n/3) for n > 1.
a(n) = 2 * A000313(n+2). - Alois P. Heinz, Jun 22 2018
E.g.f.: exp(-x)*x^2*(3 - 2*x + x^2)/(3*(1 - x)^3). - Ilya Gutkovskiy, Jun 25 2018

A345462 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) read by rows: number of distinct permutations after k steps of the "first transposition" algorithm.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 13, 4, 1, 120, 67, 23, 5, 1, 720, 411, 146, 36, 6, 1, 5040, 2921, 1067, 272, 52, 7, 1, 40320, 23633, 8800, 2311, 456, 71, 8, 1, 362880, 214551, 81055, 21723, 4419, 709, 93, 9, 1, 3628800, 2160343, 825382, 224650, 46654, 7720, 1042, 118, 10, 1

Offset: 1

Olivier Gérard, Jun 20 2021

The first transposition algorithm is: if the permutation is sorted, then exit; otherwise, exchange the first unsorted letter with the letter currently at its index. Repeat.
At each step at least 1 letter (possibly 2) is sorted.
If one counts the steps necessary to reach the identity, this gives the Stirling numbers of the first kind (reversed).

			Triangle begins:
      1;
      2,     1;
      6,     3,    1;
     24,    13,    4,    1;
    120,    67,   23,    5,   1;
    720,   411,  146,   36,   6,  1;
   5040,  2921, 1067,  272,  52,  7, 1;
  40320, 23633, 8800, 2311, 456, 71, 8, 1;
  ...

D. E. Knuth, The Art of Computer Programming, Vol. 3 / Sorting and Searching, Addison-Wesley, 1973.

Alois P. Heinz, Rows n = 1..150, flattened

Cf. A321352, A345461 (same idea for other sorting algorithms).
Cf. A180191 (second column, k=1).
Cf. A107111 a triangle with some common parts.
Cf. A143689 (diagonal T(n,n-3)).
Cf. A000142, A000166, A000255, A000274, A000313, A010027, A123513.

b:= proc(n, k) option remember; (k+1)!*
      binomial(n, k)*add((-1)^i/i!, i=0..k+1)/n
    end:
T:= proc(n, k) option remember;
     `if`(k=0, n!, T(n, k-1)-b(n, n-k+1))
    end:
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Aug 11 2021

b[n_, k_] := b[n, k] = (k+1)!*Binomial[n, k]*Sum[(-1)^i/i!, {i, 0, k+1}]/n;
T[n_, k_] := T[n, k] = If[k == 0, n!, T[n, k-1] - b[n, n-k+1]];
Table[Table[T[n, k], {k, 0, n - 1}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

T(n,0) = n!; T(n,n-3) = (3*(n-1)^2 - n + 3)/2.
From Alois P. Heinz, Aug 11 2021: (Start)
T(n,k) = T(n,k-1) - A010027(n,n-k) for k >= 1.
T(n,k) - T(n,k+1) = A123513(n,k).
T(n,0) - T(n,1) = A000255(n-1) for n >= 2.
T(n,1) - T(n,2) = A000166(n) for n >= 3.
T(n,2) - T(n,3) = A000274(n) for n >= 4.
T(n,3) - T(n,4) = A000313(n) for n >= 5. (End)

A038033 a(n) = A000166(n-1)n(n-1).

Original entry on oeis.org

6, 24, 180, 1320, 11130, 103824, 1067976, 12014640, 146845710, 1938363240, 27489515196, 416924313624, 6734931220290, 115455963776160, 2093601476474640, 40040128237577184, 805513168073611926

Offset: 3

Christian G. Bower from a sequence by Erich Friedman

Equals 6 * A000313(n+1).

Cf. A000240

a:=n->sum(n!*sum((-1)^k/k!, j=0..n), k=0..n): seq(a(n)*n, n=2..17); # Zerinvary Lajos, Dec 18 2007

E.g.f.: x^3/((1-x)^2*e^x)

cp's OEIS Frontend

A010027 Triangle read by rows: T(n,k) is the number of permutations of [n] having k consecutive ascending pairs (0 <= k <= n-1).

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A123513 Triangle read by rows: T(n,k) is the number of permutations of [n] having k small descents (n >= 1; 0 <= k <= n-1). A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) = 1.

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A000274 Number of permutations of length n with 2 consecutive ascending pairs.

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A001260 Number of permutations of length n with 4 consecutive ascending pairs.

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A001261 Number of permutations of length n with 5 consecutive ascending pairs.

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A065087 a(n) = A000166(n)*binomial(n+1,2).

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A305730 a(n) is the total displacement of all letters in all permutations of n letters with no fixed points.

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A038033 a(n) = A000166(n-1)n(n-1).