A010027 -id:A010027 - cp's OEIS frontend

A384905 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k maximal anti-runs (decreasing by more than 1).

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 6, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 7, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 21 2025

			The T(10,2) = 3 strict partitions with 2 maximal anti-runs are: (7,2,1), (5,4,1), (5,3,2).
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  2  0  0  0
  0  2  1  0  0  0
  0  3  0  1  0  0  0
  0  3  2  0  0  0  0  0
  0  4  2  0  0  0  0  0  0
  0  5  2  1  0  0  0  0  0  0
  0  6  3  0  1  0  0  0  0  0  0
  0  7  4  1  0  0  0  0  0  0  0  0
  0  9  3  3  0  0  0  0  0  0  0  0  0

Row sums are A000009.
Column k = 1 is A003114.
For subsets instead of strict integer partitions see A053538, A119900, A210034.
For runs instead of anti-runs we have A116674, for subsets A034839.
This is the strict case of A268193.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A010027, A384177, A384886, A384889, A384890.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Split[#,#1!=#2+1&]]==k&]],{n,0,10},{k,0,n}]

A087981 E.g.f.: exp(-2*x) / (1-x)^2.

Original entry on oeis.org

1, 0, 2, 4, 24, 128, 880, 6816, 60032, 589312, 6384384, 75630080, 972387328, 13483769856, 200571078656, 3185540657152, 53800242216960, 962741176500224, 18195808235880448, 362183230599856128, 7572922094360723456, 165945771111208714240, 3802923921298533384192, 90965940197460917878784, 2267151124921333646884864

Offset: 0

Gordon F. Royle, Oct 28 2003

Permanent of an (n+1) X (n+1) (+1, -1)-matrix with exactly n -1's on the diagonal and 1's everywhere else.
It is conjectured by Kräuter and Seifter that for n >= 5 a(n-1) is the maximal possible value for the permanent of a nonsingular n X n (+1, -1)-matrix. I do not know for which values of n this has been confirmed - compare A087982. - N. J. A. Sloane
The Kräuter conjecture on permanents is true (see Budrevich and Guterman). - Sergei Shteiner, Jan 17 2020
The maximal possible value for the permanent of a singular n X n (+1, -1)-matrix is obviously n!.
Degree of the "hyperdeterminant" of a multilinear polynomial on (\P^1(\C))^n, or equivalently of an element of (\C^2)^{⊗ n}: see Gelfand, Kapranov and Zelevinsky. - Eric Rains, Mar 15 2004
(-1)^n * a(n) = Polynomials in A010027 evaluated at -1. - Ralf Stephan, Dec 15 2004
a(n) is the number of n X n (-1, 0, 1)-matrices containing in every row and every column exactly one -1 and one 1 such that the main diagonal does not contain 0's. - Vladimir Shevelev, Apr 01 2010
a(n) is the number of colored permutations with no fixed points of n elements where each cycle is one of two colors. - Michael Somos, Jan 19 2011
Binomial transform is A000255. Hankel transform is A059332. - Paul Barry, Apr 11 2011
Exponential self-convolution of subfactorials (A000166). - Vladimir Reshetnikov, Oct 07 2016

			G.f. = 1 + 2*x^2 + 4*x^3 + 24*x^4 + 128*x^5 + 880*x^6 + 6816*x^7 + ...
Since a(1) = 0, then, for n = 2, we have a(2) = -(-2)^3/4 = 2; further, for n = 3, we find a(3) = (3*6/5)*2 - (-2)^4/5 = 36/5 - 16/5 = 4. - _Vladimir Shevelev_, Apr 01 2010
a(4) = 24 because there are 6 derangements with one 4-cycle with 2^1 ways to color each derangement and 3 derangements with two 2-cycles with 2^2 ways to color each derangement. - _Michael Somos_, Jan 19 2011

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 1994; see Corollary 2.10 in Chapter 14 (p. 457).

T. D. Noe, Table of n, a(n) for n = 0..100
Mikhail V. Budrevich and Alexander E. Guterman, Kräuter conjecture on permanents is true, arXiv:1810.04439 [math.CO], 2018.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Hyperdeterminants, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.10 (p. 246).
Arnold R. Kräuter, Permanenten - Ein kurzer Überblick, Séminaire Lotharingien de Combinatoire, B09b (1983), 34 pp.
Arnold R. Kräuter and Norbert Seifter, Some properties of the permanent of (1,-1)-matrices, Linear and Multilinear Algebra 15 (1984), 207-223.
Norbert Seifter, Upper bounds for permanents of (1,-1)-matrices, Israel J. Math. 48 (1984), 69-78.
Edward Tzu-Hsia Wang, On permanents of (1,-1)-matrices, Israel J. Math. 18 (1974), 353-361.
Wikipedia, Hyperdeterminant
Index entries for sequences related to binary matrices

Cf. A087982, A087983, A176097.

Cf. A000007, A000023, A000166, A024000, A137775. - Michael Somos, Jan 19 2011

seq(simplify(KummerU(-n, -n-1, -2)), n = 0..24); # Peter Luschny, May 10 2022

Range[0, 20]! CoefficientList[Series[Exp[-2 x]/(1 - x)^2, {x, 0, 20}], x]
Table[(-2)^n HypergeometricPFQ[{2, -n}, {}, 1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( -2 * x + x * O(x^n) ) / ( 1 - x )^2, n ) )} /* Michael Somos, Jan 19 2011 */

Krauter and Seifter prove that the permanent of an n X n {-1, 1} matrix is divisible by 2^{n - [log_2(n)] - 1}.
Let c(n) be the permanent of the {-1, +1}-matrix of order n X n with n diagonal -1's only. Let a(n) be the permanent of the {-1, +1}-matrix of order (n+1) X (n+1) with n diagonal -1's only. Then by expanding along the first row (like determinant, but with no sign) we get c(n+1) = -c(n) + n a(n-1), a(n) = c(n) + n a(n-1), with c(2) = 2, a(2) = 2. {c(n)} has e.g.f. exp(-2x)/(1-x), see A000023. Also a(n) = c(n+1) + 2*c(n).
The following 4 formulas hold: a(n) = Sum_{k = 0..n} C(n, k)*D_k*D_{n-k}, where D_n = A000166(n); a(n) = n!*Sum_{j = 0..n} (n+1-j)*(-2)^j/j!; a(0) = 1, a(1) = 0 and, for n > 0, a(n+1) = n*(a(n) + 2*a(n-1)); a(0) = 1 and, for n > 0, a(n) = (n*(n+3)/(n+2))*a(n-1) - (-2)^(n+1)/(n+2). - Vladimir Shevelev, Apr 01 2010 [edited by Michael Somos, Jan 19 2011]
G.f.: 1/(1-2x^2/(1-2x-6x^2/(1-4x-12x^2/(1-6x-20x^2/(1-.../(1-2n*x-(n+1)(n+2)x^2/(1-... (continued fraction). - Paul Barry, Apr 11 2011
E.g.f.: 1/U(0) where U(k)=  1 - 2*x/( 1 + x/(2 - x - 4/( 2 + x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: 1/Q(0) where Q(k) = 1 - 2*k*x - x^2*(k + 1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 10 2013
G.f.: S(x)/x - 1/x = G(0)/x - 1/x, where S(x) = sum(k >= 0, k!*(x/(1+2*x))^k  ), G(k) = 1 + (2*k + 1)*x/( 1+2*x - 2*x*(1+2*x)*(k+1)/(2*x*(k+1) + (1+2*x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) = (-2)^n*hypergeom([2, -n], [], 1/2) = 4*(-2)^n*(1 - 2*hypergeom([1, -n-3], [], 1/2))/(n^2+3*n+2) = (4*(-2)^n + Gamma(n+4, -2)*exp(-2))/(n^2+3*n+2). - Vladimir Reshetnikov, Oct 07 2016
a(n) ~ sqrt(2*Pi) * n^(n+3/2) / exp(n+2). - Vaclav Kotesovec, Oct 08 2016
a(n) = KummerU(-n, -n - 1, -2). - Peter Luschny, May 10 2022

More terms from Jaap Spies, Oct 28 2003
Further terms from Gordon F. Royle, Oct 29 2003
Definition via e.g.f. from Eric Rains, Mar 15 2004
Changed the offset and terms to correspond to e.g.f, Michael Somos, Jan 19 2011

A243815 Number of length n words on alphabet {0,1} such that the length of every maximal block of 0's (runs) is the same.

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 39, 62, 97, 151, 233, 360, 557, 864, 1344, 2099, 3290, 5176, 8169, 12931, 20524, 32654, 52060, 83149, 133012, 213069, 341718, 548614, 881572, 1417722, 2281517, 3673830, 5918958, 9540577, 15384490, 24817031, 40045768, 64637963, 104358789

Offset: 0

Geoffrey Critzer, Jun 11 2014

nonn

Number of terms of A164710 with exactly n+1 binary digits. - Robert Israel, Nov 09 2015
From Gus Wiseman, Jun 23 2025: (Start)
This is the number of subsets of {1..n} with all equal lengths of runs of consecutive elements increasing by 1. For example, the runs of S = {1,2,5,6,8,9} are ((1,2),(5,6),(8,9)), with lengths (2,2,2), so S is counted under a(9). The a(0) = 1 through a(4) = 14 subsets are:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}
(End)

			0110 is a "good" word because the length of both its runs of 0's is 1.
Words of the form 11...1 are good words because the condition is vacuously satisfied.
a(5) = 24 because there are 32 length 5 binary words but we do not count: 00010, 00101, 00110, 01000, 01001, 01100, 10010, 10100.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A164710.
These subsets are ranked by A164707, complement A164708.
For distinct instead of equal lengths we have A384175, complement A384176.
For anti-runs instead of runs we have A384889, for partitions A384888.
For permutations instead of subsets we have A384892, distinct instead of equal A384891.
For partitions instead of subsets we have A384904, strict A384886.
The complement is counted by A385214.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A049988 counts partitions with equal run-lengths, distinct A325325.
A329738 counts compositions with equal run-lengths, distinct A329739.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
Cf. A010027, A044813, A069010, A268193, A383013, A384177, A384893.

a:= n-> 1 + add(add((d-> binomial(d+j, d))(n-(i*j-1))
          , j=1..iquo(n+1, i)), i=2..n+1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 11 2014

nn=30;Prepend[Map[Total,Transpose[Table[Drop[CoefficientList[Series[ (1+x^k)/(1-x-x^(k+1))-1/(1-x),{x,0,nn}],x],1],{k,1,nn}]]],0]+1
Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}] (* Gus Wiseman, Jun 23 2025 *)

A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset: 0

Wouter Meeussen, May 23 2001

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012
Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012
A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x).  See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012
The matrix inverse starts
    1;
   -1,   1;
   -1,  -1,   1;
    1,  -2,  -1,  1;
    3,   1,  -3, -1,  1;
    1,   6,   1, -4, -1,  1;
   -7,   4,  10,  1, -5, -1,  1;
  -13, -13,   8, 15,  1, -6, -1,  1;
    3, -31, -23, 13, 21,  1, -7, -1, 1; - R. J. Mathar, Mar 15 2013
Also appears to be the number of subsets of {1..n} containing n with k maximal anti-runs of consecutive elements increasing by more than 1. For example, the subset {1,3,6,7,11,12} has maximal anti-runs ((1,3,6),(7,11),(12)) so is counted under a(12,3). For runs instead of anti-runs we get A202064. - Gus Wiseman, Jun 26 2025

			n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened
R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Column k = 1 is A000045.
Row sums are A000079.
Column k = 2 is A010049.
For runs instead of anti-runs we have A202064.
For integer partitions see A268193, strict A384905, runs A116674.
A034839 counts subsets by number of maximal runs.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs.
Cf. A000071, A001629, A010027, A208342, A384177, A384879, A384889, A384890.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j,k)*Binomial(n-j,j-k)) ))); # G. C. Greubel, May 16 2019

[[(&+[Binomial(j,k)*Binomial(n-j,j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n,m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k,0,n}], {n,0,12}, {m,0,n}]

{T(n,k) = sum(j=0,n, binomial(j,k)*binomial(n-j,j-k))}; \\ G. C. Greubel, May 16 2019

[[sum(binomial(j,k)*binomial(n-j,j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019

From Philippe Deléham, Mar 05 2012: (Start)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.
G.f.: 1/(1-(1+y)*x-(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 4, 8, 14, 23, 37, 59, 93, 146, 230, 365, 584, 940, 1517, 2450, 3959, 6404, 10373, 16822, 27298, 44297, 71843, 116429, 188550, 305200, 493930, 799422, 1294108, 2095291, 3392736, 5493168, 8892148, 14390372, 23282110, 37660759, 60914308, 98528312, 159386110

Offset: 0

Gus Wiseman, Jun 18 2025

nonn

			The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), with lengths (2,2,2), so is counted under a(12).
The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}

Christian Sievers, Table of n, a(n) for n = 0..1000

For runs instead of anti-runs we have A243815, distinct A384175, complement A384176.
For distinct instead or equal lengths we have A384177, ranks A384879.
For partitions instead of subsets we have A384888.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A047966 counts uniform partitions (equal multiplicities), ranks A072774.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A047993, A356606, A384880, A384886, A384890.

Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]

lista(n)=Vec(sum(i=1,(n+1)\2,1/(1-x^(2*i-1)/(1-x)^(i-1))-1,1-x+O(x*x^n))/(1-x)^2) \\ Christian Sievers, Jun 20 2025

G.f.: ( Sum_{i>=1} (1/(1-x^(2*i-1)/(1-x)^(i-1))-1) + 1-x ) / (1-x)^2. - Christian Sievers, Jun 21 2025

a(21) and beyond from Christian Sievers, Jun 20 2025

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

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (x/(1-x)^2, x^2/(1-x)^2).
Mirror image of triangle in A119900.
A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011
From Gus Wiseman, Jul 07 2025: (Start)
Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:
  {5}          {1,5}      {1,3,5}
  {4,5}        {2,5}
  {3,4,5}      {3,5}
  {2,3,4,5}    {1,2,5}
  {1,2,3,4,5}  {1,4,5}
               {2,3,5}
               {2,4,5}
               {1,2,3,5}
               {1,2,4,5}
               {1,3,4,5}
For anti-runs instead of runs we have A053538.
Without requiring n see A210039, A202023, reverse A098158, A109446.
(End)

			Triangle begins :
1
2, 0
3, 1, 0
4, 4, 0, 0
5, 10, 1, 0, 0
6, 20, 6, 0, 0, 0
7, 35, 21, 1, 0, 0, 0
8, 56, 56, 8, 0, 0, 0, 0

Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.
Column k = 1 is A000027.
Row sums are A000079.
Column k = 2 is A000292.
Without zeros we have A034867.
Last nonzero term in each row appears to be A124625.
A034839 counts subsets by number of maximal runs, for anti-runs A384893.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.
Cf. A098158, A109446, A202023, A210039.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* Gus Wiseman, Jul 07 2025 *)

G.f.: 1/((1-x)^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.
T(n,k) = binomial(n+1,2k+1).
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

A240796 Total number of occurrences of the pattern 1<2 in all preferential arrangements (or ordered partitions) of n elements.

Original entry on oeis.org

0, 1, 15, 186, 2330, 31065, 447405, 6979588, 117745668, 2141106795, 41810587775, 873474855726, 19451904450654, 460209050303821, 11531197020389025, 305122289460210120, 8503747639606509128, 249020038061419770783, 7645072502094118876755, 245564189847880300238290

Offset: 1

N. J. A. Sloane, Apr 13 2014

nonn

There are A000670(n) preferential arrangements of n elements - see A000670, A240763.
The number that avoid the pattern 1<2  is 2^(n-1).
The total number of occurrences of the pattern 1<2 in all permutations on n elements is (n-1)*(n-1)! (cf. A010027, A001563).

			The 13 preferential arrangements on 3 points and the number of times the pattern 1<2 occurs are:
1<2<3, 3
1<3<2, 2
2<1<3, 2
2<3<1, 1
3<1<2, 1
3<2<1, 0
1=2<3, 2
1=3<2, 1
2=3<1, 0
1<2=3, 2
2<1=3, 1
3<1=2, 0
1=2=3, 0,
for a total of a(3) = 15.

Alois P. Heinz, Table of n, a(n) for n = 1..420

Cf. A000670, A240763, A240796-A240800, A010027, A001563, A264082, A381299.

b:= proc(n, t) option remember; `if`(n=0, [1, 0], add((p-> p+
      [0, p[1]*j*t/2])(b(n-j, t+j))*binomial(n, j), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 08 2014

b[n_, t_] := b[n, t] = If[n == 0, {1, 0}, Sum[Function[{p}, p + {0, p[[1]]*j*t/2}][b[n - j, t + j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Jun 08 2015, after Alois P. Heinz *)

a(n) ~ n! * n^2 / (8 * (log(2))^(n+1)). - Vaclav Kotesovec, May 03 2015

a(n) = Sum_{k=0..binomial(n,2)} k * A381299(n,k). - Alois P. Heinz, Feb 22 2025

a(8)-a(20) from Alois P. Heinz, Dec 08 2014

cp's OEIS Frontend

A384905 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k maximal anti-runs (decreasing by more than 1).

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A087981 E.g.f.: exp(-2*x) / (1-x)^2.

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A243815 Number of length n words on alphabet {0,1} such that the length of every maximal block of 0's (runs) is the same.

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A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

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A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 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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