A001260 -id:A001260 - 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

A000313 Number of permutations of length n with 3 consecutive ascending pairs.

Original entry on oeis.org

0, 0, 0, 1, 4, 30, 220, 1855, 17304, 177996, 2002440, 24474285, 323060540, 4581585866, 69487385604, 1122488536715, 19242660629360, 348933579412440, 6673354706262864, 134252194678935321, 2834212998777523380, 62651024183503148470, 1447238658638922729580

Offset: 1

N. J. A. Sloane

Temporary remark: there may be some issues with respect to the offset of this sequence in the formula and program sections. - Joerg Arndt, Nov 16 2014

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).

Todd Silvestri, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe)

Cf. A010027, A000255, A000166, A000274, A001260, A001261.

A diagonal in triangle A010027.

series(hypergeom([2,4],[],x/(x+1))/(x+1)^4, x=0, 30); # Mark van Hoeij, Nov 07 2011
a := n -> simplify(hypergeom([4-n,2],[],1))*(-1)^n*(n-1)*(n-2)*(n-3)/6: seq(a(n), n=1..23); # Peter Luschny, Nov 19 2014

Table[(n*(n + 1)!/6)*Sum[(-1)^k/k!, {k, 0, n}], {n, -1, 25}] (* T. D. Noe, Jun 19 2012 *)
a[1]:=0; a[n_Integer/;n>=2]:=(n-2) (n-1) Subfactorial[n-2]/6 (* Todd Silvestri, Nov 15 2014 *)

a = lambda n: (n-2)*(n-1)*sloane.A000166(n-2)/6 if n>2 else 0
[a(n) for n in range(1,24)] # Peter Luschny, Nov 19 2014

a(n) = (n*(n+1)!/6)*sum((-1)^k/k!, k=0..n).
a(n) = A065087(n+2)/3. - Zerinvary Lajos, May 25 2007
E.g.f.: x^3/3!*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
a(n) = round( (exp(-1)*(n+1)!+(-1)^n)*n/6 ). - Mark van Hoeij, Oct 25 2011
G.f.: hypergeom([2, 4],[],x/(x+1))/(x+1)^4. - Mark van Hoeij, Nov 07 2011
a(1) = 0, a(n) = (n-2)*(n-1)*(!(n-2))/6 = (n-2)*(n-1)*A000166(n-2)/6, for n >= 2. - Todd Silvestri, Nov 15 2014
a(n) = hypergeom([4-n,2],[],1)*(-1)^n*A000292(n-3). - Peter Luschny, Nov 19 2014
D-finite with recurrence (-n+4)*a(n) +(n-1)*(n-4)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

More terms from Vladeta Jovovic, Jan 03 2003
Formula added by Sean A. Irvine, Nov 11 2010
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

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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A000313 Number of permutations of length n with 3 consecutive ascending pairs.

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

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