A260665 -id:A260665 - cp's OEIS frontend

A001754 Lah numbers: a(n) = n!*binomial(n-1,2)/6.

0, 0, 1, 12, 120, 1200, 12600, 141120, 1693440, 21772800, 299376000, 4390848000, 68497228800, 1133317785600, 19833061248000, 366148823040000, 7113748561920000, 145120470663168000, 3101950060425216000, 69337707233034240000, 1617879835437465600000

Offset: 1

N. J. A. Sloane

a(n+1) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0)^2, i.e., the sum of the squares of the positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters, Oct 25 2006

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
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..300

Column 3 of A008297.
Column m=3 of unsigned triangle A111596.
Cf. A005990, A053495, A260665.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n)*Binomial(n-1, 2)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011

Maple
```
[seq(n!*binomial(n-1,2)/6, n=1..40)];
```

Table[(n-2)*(n-1)*n!/12, {n, 21}] (* Arkadiusz Wesolowski, Nov 26 2012 *)
With[{nn=30},CoefficientList[Series[(x/(1-x))^3/6,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2017 *)

[factorial(n-1)*binomial(n,3)/2 for n in (1..30)] # G. C. Greubel, May 10 2021

E.g.f.: ((x/(1-x))^3)/3!.
If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i)*x^(k-j)) then a(n+1) = (-1)^n*f(n,2,-4), n >= 2. - Milan Janjic, Mar 01 2009
a(n) = Sum_{k>=1} k * A260665(n,k). - Alois P. Heinz, Nov 14 2015
D-finite with recurrence (-n+5)*a(n) + (n-2)*(n-3)*a(n-1) = 0, n >= 4. - R. J. Mathar, Jan 06 2021
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 6*(gamma - Ei(1)) + 9, where gamma = A001620 and Ei(1) = A091725.
Sum_{n>=3} (-1)^(n+1)/a(n) = 18*(gamma - Ei(-1)) - 12/e - 9, where Ei(-1) = -A099285 and e = A001113. (End)

A263776 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 8, 2, 42, 45, 25, 7, 1, 132, 220, 198, 112, 44, 12, 2, 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1, 1430, 4368, 7280, 8400, 7460, 5392, 3262, 1664, 716, 256, 74, 16, 2, 4862, 18564, 38556, 56100, 63648, 59670, 47802, 33338, 20466, 11115

Offset: 0

Christian Stump, Oct 26 2015

Row sums give A000142.
First column gives A000108.
Also the number of permutations of [n] with k crossings (see Corteel, Proposition 4).
Also the number of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2). - Alois P. Heinz, Nov 14 2015

			Triangle begins:
0 :   1;
1 :   1;
2 :   2;
3 :   5,    1;
4 :  14,    8,    2;
5 :  42,   45,   25,    7,   1;
6 : 132,  220,  198,  112,  44,  12,   2;
7 : 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1;
...

Alois P. Heinz, Rows n = 0..50, flattened
A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001.
S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.
FindStat - Combinatorial Statistic Finder, The number of nestings of a permutation, The number of crossings of a permutation
R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021.

Columns k=0-10 give: A000108, A002696, A094218, A094219, A120812, A120813, A120814, A120815, A120816, A264496, A264497.

Cf. A000142, A001754, A002620, A260665, A260670, A287328, A291722.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
       add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 14 2015

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^(j-1)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[ T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A001754(n).

T(n,n) = A287328(n). - Alois P. Heinz, Aug 31 2017

More terms from Alois P. Heinz, Oct 26 2015

A260670 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 23-1; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 15, 6, 3, 52, 32, 23, 10, 3, 203, 171, 152, 98, 62, 22, 11, 1, 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4, 4140, 5444, 6460, 6082, 5513, 4302, 3328, 2141, 1393, 780, 432, 187, 88, 24, 6, 21147, 32919, 43626, 46508, 46880, 41979, 36774

Offset: 0

Alois P. Heinz, Nov 14 2015

Patterns 1-32, 3-12, 21-3 give the same sequence.

			T(3,1) = 1: 231.
T(4,1) = 6: 1342, 2314, 2413, 2431, 3241, 4231.
T(4,2) = 3: 2341, 3412, 3421.
T(5,2) = 23: 13452, 14523, 14532, 23415, 23514, 23541, 24351, 25341, 32451, 34125, 34152, 34215, 35124, 35142, 35214, 35412, 35421, 42351, 43512, 43521, 52341, 53412, 53421.
T(5,3) = 10: 23451, 24513, 24531, 34251, 35241, 45123, 45132, 45213, 45312, 45321.
T(5,4) = 3: 34512, 34521, 45231.
Triangle T(n,k) begins:
0 :   1;
1 :   1;
2 :   2;
3 :   5,   1;
4 :  15,   6,   3;
5 :  52,  32,  23,  10,   3;
6 : 203, 171, 152,  98,  62,  22,  11,   1;
7 : 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4;

Alois P. Heinz, Rows n = 0..50, flattened
A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001

Columns k=0-10 give: A000110, A264460, A264461, A264462, A264463, A264464, A264465, A264466, A264467, A264468, A264469.
Row sums give A000142.
Cf. A001754, A125811, A260665, A263776.

b:= proc(u, o) option remember;
     `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
       add(expand(b(u+j-1, o-j)*x^u), j=1..o))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n,0)):
seq(T(n), n=0..10);

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^u], {j, 1, o}]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 16 2017, after Alois P. Heinz *)

Sum_{k>0} k * T(n,k) = A001754(n).

A092923 Number of permutations containing exactly one occurrence of the pattern #, with # one of {1-23, 3-21, 12-3, 32-1}.

Original entry on oeis.org

1, 7, 39, 211, 1168, 6728, 40561, 256297, 1696707, 11752973, 85047284, 641782220, 5041634549, 41160207335, 348664792199, 3059885806071, 27781291314396, 260599397789924, 2522492941426381, 25166308238897929, 258507111338795491, 2731176458973448817

Offset: 3

Ralf Stephan, Apr 18 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..500
A. Claesson and T. Mansour, Counting patterns of type (1,2) or (2,1), arXiv:math/0110036 [math.CO], 2001.

Column k=1 of A260665.

a[n_ /; n<3] = 0; a[n_] := a[n] = 2 a[n-1] + Sum[Binomial[n-2, k] (a[k+1] + BellB[k+1]), {k, 0, n-3}];
Table[a[n], {n, 3, 24}] (* Jean-François Alcover, Aug 19 2018 *)

a(n)=if(n<1,0,2*a(n-1)+sum(k=0,n-3,binomial(n-2,k)*(a(k+1)+polcoeff(serlaplace(exp(exp(x)-1)),k+1))))

G.f.: Sum_{n>=1} (x/(1-n*x)) * Sum_{k>=0} k*x^(k+n)/Product_{l=1..k+n} (1-l*x).

Recurrence: a(n) = 2a(n-1) + Sum_{k=0..n-3} C(n-2, k)*(a(k+1) + B(k+1)), with B(n) the Bell numbers A000110(n).

A264451 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the generalized pattern 12-3.

Original entry on oeis.org

1, 13, 112, 843, 6089, 43887, 321357, 2411686, 18631631, 148490575, 1221894598, 10382648734, 91073271181, 824221683639, 7690752545310, 73932347642395, 731636916804531, 7447376286108705, 77913394499688645, 837113596053971008, 9229808681562243113

Offset: 4

Alois P. Heinz, Nov 14 2015

nonn

			a(4) = 1: 1243.
a(5) = 13: 13254, 13524, 13542, 14235, 21354, 23154, 23514, 23541, 24135, 31254, 34125, 41253, 51243.

Alois P. Heinz, Table of n, a(n) for n = 4..500

Column k=2 of A260665.

b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
       b(u-j, o+j-1), j=1..u)+add(convert(series(
       b(u+j-1, o-j)*x^(o-j), x, 3), polynom), j=1..o))
    end:
a:= n-> coeff(b(n, 0), x, 2):
seq(a(n), n=4..25);

A264452 Number of permutations of [n] with exactly three (possibly overlapping) occurrences of the generalized pattern 12-3.

Original entry on oeis.org

1, 12, 103, 811, 6273, 48806, 386041, 3122069, 25900188, 220791812, 1935811756, 17461471292, 162038542377, 1546528485770, 15174625184321, 152994813139537, 1584120732683571, 16834244135840106, 183496533502593453, 2050337555698723711, 23470542944212951050

Offset: 4

Alois P. Heinz, Nov 14 2015

nonn

			a(4) = 1: 1234.
a(5) = 12: 12534, 12543, 13245, 13425, 13452, 21345, 23145, 23415, 23451, 31245, 41235, 51234.

Alois P. Heinz, Table of n, a(n) for n = 4..500

Column k=3 of A260665.

b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
       b(u-j, o+j-1), j=1..u)+add(convert(series(
       b(u+j-1, o-j)*x^(o-j), x, 4), polynom), j=1..o))
    end:
a:= n-> coeff(b(n, 0), x, 3):
seq(a(n), n=4..25);

A264453 Number of permutations of [n] with exactly four (possibly overlapping) occurrences of the generalized pattern 12-3.

Original entry on oeis.org

2, 41, 492, 4851, 44291, 394154, 3502159, 31450283, 287335089, 2680535274, 25586566853, 250169461829, 2506738712869, 25745302689807, 270994937686871, 2922760610595872, 32288420619988347, 365214362676429903, 4227719744153328064, 50063651356822568135

Offset: 5

Alois P. Heinz, Nov 14 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..500

Column k=4 of A260665.

A264454 Number of permutations of [n] with exactly five (possibly overlapping) occurrences of the generalized pattern 12-3.

Original entry on oeis.org

1, 24, 337, 3798, 38795, 379611, 3659389, 35291284, 343556139, 3394140548, 34141603945, 350367304972, 3672490264985, 39344027488644, 430933818393770, 4825995839698429, 55254816421477438, 646659542446548212, 7733703865241542989, 94486241595833611364

Offset: 5

Alois P. Heinz, Nov 14 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..500

Column k=5 of A260665.

A264455 Number of permutations of [n] with exactly six (possibly overlapping) occurrences of the generalized pattern 12-3.

Original entry on oeis.org

1, 17, 238, 2956, 33343, 355182, 3681762, 37839385, 389940538, 4056942189, 42796503464, 458995025669, 5013561835833, 55832699192796, 634322118799677, 7354538207075076, 87032355679553323, 1051183375285792257, 12956782873283747998, 162950022525866642259

Offset: 5

Alois P. Heinz, Nov 14 2015

nonn

			a(5) = 1: 12345.
a(6) = 17: 124365, 124635, 124653, 125346, 132456, 134256, 134526, 134562, 213456, 231456, 234156, 234516, 234561, 312456, 412356, 512346, 612345.

Alois P. Heinz, Table of n, a(n) for n = 5..500

Column k=6 of A260665.

b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
       b(u-j, o+j-1), j=1..u)+add(convert(series(
       b(u+j-1, o-j)*x^(o-j), x, 7), polynom), j=1..o))
    end:
a:= n-> coeff(b(n, 0), x, 6):
seq(a(n), n=5..25);

A264456 Number of permutations of [n] with exactly seven (possibly overlapping) occurrences of the generalized pattern 12-3.

Original entry on oeis.org

5, 122, 1960, 25922, 308912, 3488684, 38438284, 420224213, 4604712333, 50891058160, 569534211553, 6470640729869, 74755213239764, 879145839345147, 10531462769901577, 128554339635477023, 1599307831822386125, 20278763016658278490, 262052769447634167173

Offset: 6

Alois P. Heinz, Nov 14 2015

nonn

			a(6) = 5: 123645, 123654, 124356, 124536, 124563.
a(7) = 122: 1253746, 1253764, 1254376, ..., 7124356, 7124536, 7124563.

Alois P. Heinz, Table of n, a(n) for n = 6..500

Column k=7 of A260665.

b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
       b(u-j, o+j-1), j=1..u)+add(convert(series(
       b(u+j-1, o-j)*x^(o-j), x, 8), polynom), j=1..o))
    end:
a:= n-> coeff(b(n, 0), x, 7):
seq(a(n), n=6..25);

cp's OEIS Frontend

A001754 Lah numbers: a(n) = n!*binomial(n-1,2)/6.

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A263776 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.

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A260670 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 23-1; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows.

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A092923 Number of permutations containing exactly one occurrence of the pattern #, with # one of {1-23, 3-21, 12-3, 32-1}.

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A264451 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the generalized pattern 12-3.

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A264452 Number of permutations of [n] with exactly three (possibly overlapping) occurrences of the generalized pattern 12-3.

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A264453 Number of permutations of [n] with exactly four (possibly overlapping) occurrences of the generalized pattern 12-3.

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A264454 Number of permutations of [n] with exactly five (possibly overlapping) occurrences of the generalized pattern 12-3.

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