A000680 -id:A000680 - cp's OEIS frontend

A334776 Total number of peaks in all permutations of 2 indistinguishable copies of 1..n.

0, 3, 105, 4620, 283500, 23700600, 2610808200, 367783416000, 64607286744000, 13859305059600000, 3567385122341040000, 1085582734152396480000, 385634331725066424000000, 158175715893528308976000000, 74203019661816956710800000000, 39481403043334753112451840000000

Offset: 1

Andrew Howroyd, May 12 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000680, A334774, A334775, A334777.

\\ PeaksBySig defined in A334774.
a(n)={my(u=PeaksBySig(vector(n,i,2), [0..n-1])); sum(k=1, #u, (k-1)*u[k])}

a(n) = Sum_{k=1..n} (k-1)*A334774(n,k).

a(n) = A334777(n) - A000680(n).

A334777 Total number of local maxima in all permutations of 2 indistinguishable copies of 1..n.

Original entry on oeis.org

1, 9, 195, 7140, 396900, 31185000, 3291888600, 449513064000, 77111922888000, 16235185926960000, 4116213602701200000, 1237059394731800640000, 434864246413372776000000, 176784623645708110032000000, 82297894534015170170160000000, 43496460979945066988294400000000

Offset: 1

Andrew Howroyd, May 12 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000680, A334774, A334775, A334776.

\\ PeaksBySig defined in A334774.
a(n)={my(u=PeaksBySig(vector(n,i,2), [0..n-1])); sum(k=1, #u, k*u[k])}

a(n) = Sum_{k=1..n} k*A334774(n,k).

a(n) = A334776(n) + A000680(n).

A361651 Number T(n,k) of permutations p of [n] such that p(i), p(i+k), p(i+2k),... form an up-down sequence for i in [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 3, 6, 0, 5, 6, 12, 24, 0, 16, 20, 30, 60, 120, 0, 61, 80, 90, 180, 360, 720, 0, 272, 350, 420, 630, 1260, 2520, 5040, 0, 1385, 1750, 2240, 2520, 5040, 10080, 20160, 40320, 0, 7936, 10080, 13440, 15120, 22680, 45360, 90720, 181440, 362880

Offset: 0

Alois P. Heinz, Mar 19 2023

Number T(n,k) of permutations p of [n] such that p(i) < p(i+k) > p(i+2k) < ... for i <= k.

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = n! for k>=n.

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,    2;
  0,    2,    3,    6;
  0,    5,    6,   12,   24;
  0,   16,   20,   30,   60,  120;
  0,   61,   80,   90,  180,  360,   720;
  0,  272,  350,  420,  630, 1260,  2520,  5040;
  0, 1385, 1750, 2240, 2520, 5040, 10080, 20160, 40320;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Columns k=0-3 give: A000007, A000111, A361648, A367336.
Main diagonal gives A000142.
T(2n,n) gives A000680.
Cf. A248686, A333706.

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
T:= (n, k)-> `if`(n=0, 1, `if`(k=0, 0, (l-> mul(b(s, 0), s=l)*
    combinat[multinomial](n, l[]))([floor((n+i)/k)$i=0..k-1]))):
seq(seq(T(n, k), k=0..n), n=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
T[n_, k_] := If[n == 0, 1, If[k == 0, 0, Function[l, Product[b[s, 0], {s, l}]*multinomial[n, l]][Table[Floor[(n+i)/k], {i, 0, k-1}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Nov 22 2023, after Alois P. Heinz *)

A089975 Array read by ascending antidiagonals: T(n,k) is the number of n-letter words from a k-letter alphabet such that no letter appears more than twice.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 4, 3, 1, 0, 0, 6, 9, 4, 1, 0, 0, 6, 24, 16, 5, 1, 0, 0, 0, 54, 60, 25, 6, 1, 0, 0, 0, 90, 204, 120, 36, 7, 1, 0, 0, 0, 90, 600, 540, 210, 49, 8, 1, 0, 0, 0, 0, 1440, 2220, 1170, 336, 64, 9, 1, 0, 0, 0, 0, 2520, 8100, 6120, 2226, 504, 81, 10, 1

Offset: 0

Paul Boddington, Nov 17 2003

			Array begins:
  1, 1, 1,  1,    1,     1,      1,      1,       1,       1,       1, ...
  0, 1, 2,  3,    4,     5,      6,      7,       8,       9,      10, ...
  0, 1, 4,  9,   16,    25,     36,     49,      64,      81,     100, ...
  0, 0, 6, 24,   60,   120,    210,    336,     504,     720,     990, ...
  0, 0, 6, 54,  204,   540,   1170,   2226,    3864,    6264,    9630, ...
  0, 0, 0, 90,  600,  2220,   6120,  14070,   28560,   52920,   91440, ...
  0, 0, 0, 90, 1440,  8100,  29520,  83790,  201600,  430920,  842400, ...
  0, 0, 0,  0, 2520, 25200, 128520, 463680, 1345680, 3356640, 7484400, ...
  ... - _Robert FERREOL_, Nov 03 2017

Dennis Walsh, Width-Restricted Finite Functions

T(1, k) = A001477(k); T(2, k) = A000290(k); T(3, k) = A007531(k); T(n, n) = A012244(n); T(n, n+1) = A036774(n); T(n, n+2) = A003692(n+1); T(2*n, n) = A000680(n); sum(T(n, k), n=0..2*k) = A003011(k); sum(T(r, n-r), r=0..n) = A089976(n).

See A141765 for an irregular triangle version : T(n,k)=A141765(k,n) for n <= 2k.

T:=(n,k)->add(n!*k!/(n-2*i)!/i!/(k-n+i)!/2^i,i=max(0,n-k)..n/2):
or
T:=proc(n,k) option remember :if n=0 then 1 elif n=1 then k elif k=0 then 0 else T(n, k-1)+n*T(n-1, k-1)+binomial(n,2)*T(n-2, k-1) fi end:
or
T:=(n,k)-> n!*coeff((1 + x + x^2/2)^k, x,n):
seq(seq(T(n-k,k),k=0..n),n=0..20);
# Robert FERREOL, Nov 07 2017

T[n_, k_] := Sum[n!*k!/(2^i*(n - 2 i)!*(k - n + i)!*i!), {i, Max[0, n - k], Floor[n/2]}];
Table[T[n-k , k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 05 2017, after Robert FERREOL *)

from math import factorial as f
def T(n,k):
    return sum(f(n)*f(k)//f(n-2*i)//f(i)//f(k-n+i)//2**i for i in range(max(0,n-k),n//2+1))
[T(n-k,k) for n in range(21) for k in range(n+1)]
# Robert FERREOL, Oct 17 2017

T(n, k) = T(n, k-1) + n*T(n-1, k-1) + binomial(n, 2)*T(n-2, k-1) for n >= 2 and k >= 1.
T(n, k) = Sum_{i=max(0,n-k)..floor(n/2)} n!*k!/(2^i*(n-2*i)!*(k-n+i)!*i!). - Robert FERREOL, Oct 30 2017
T(n,k) = (-1)^n*n!*2^(-n/2)*GegenbauerC(n, -k, 1/sqrt(2)) for k >= n. - Robert Israel, Nov 08 2017
G.f.: Sum({n>=0} T(n,k)x^n)=n!(1 + x + x^2/2)^k. See Walsh link. - Robert FERREOL, Nov 14 2017

A117414 An Euler triangle.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 48, 12, 1, 0, 1088, 272, 24, 1, 0, 39680, 9920, 880, 40, 1, 0, 2122752, 530688, 47104, 2160, 60, 1, 0, 156577792, 39144448, 3474688, 159488, 4480, 84, 1, 0, 15230058496, 3807514624, 337979392, 15514880, 436352, 8288, 112, 1

Offset: 0

Paul Barry, Mar 13 2006

Conjecture: row sums are the Euler numbers A000364. Second column is A024255. Third column is A117415.

Here, w = w_1,w_2,...,w_(2n) is an alternating permutation if w_1 < w_2 > w_3 < ... > w_(2n-1) < w_2n. An alternating permutation is cyclically alternating if w_1 < w_(2n). Define the cyclically alternating decomposition of w in the following manner: From the set {w_2,w_4,w_6,...,w_(2n)} find the largest i such that w_(2i) > w_1. Then w_1,w_2,...,w_(2i) is the first component in the cyclically alternating decomposition of w. Repeat the process with the set {w_(2i+1),w_(2i+2),...,w_(2n)} to find the successive components. Conjecture: T(n,k) is the number of alternating permutations of [2n] with exactly k cyclically alternating components. - Geoffrey Critzer, Apr 26 2023

			Triangle begins:
  1;
  0,       1;
  0,       4,      1;
  0,      48,     12,     1;
  0,    1088,    272,    24,    1;
  0,   39680,   9920,   880,   40,  1;
  0, 2122752, 530688, 47104, 2160, 60, 1;
  ...

N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003, page 9.

Cf. A000364, A024255, A117415, A086646, A000680.

nn = 6; B[n_] := (2 n)!/2^n; e[z_] := Sum[z^n/B[n], {n, 0, nn}];
Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[
Series[e[(u - 1) z] 1/e[-z], {z, 0, nn}], {z, u}]] // Grid (* Geoffrey Critzer, Apr 26 2023 *)

From Geoffrey Critzer, Apr 26 2023: (Start)
Sum_{n>=0} Sum_{k=0..n} T(n,k)*u^k*z^n/A000680(n) = E((u-1)*z)/E(-z) Where E(z) = Sum_{n>=0} z^n/A000680(n).
Sum_{k=0..n} T(n,k)*k = A086646(n,1). (End)

A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.

Original entry on oeis.org

18, 576, 46200, 7484400, 2137544640, 981562982400, 678245967907200, 670873729125600000, 913601739437346960000, 1660189302321994373529600, 3923769742187622047360640000, 11805614186177306251101945600000, 44403795869109177300313209696000000

Offset: 3

Jonathan Vos Post, Nov 25 2007

Array T(n,k) = k-th polygorial(n,k) begins:
k  |  polygorial(n,k)
| 1 1  3  18   180    2700     56700     1587600      57153600
| 1 1  4  36   576   14400    518400    25401600    1625702400
| 1 1  5  60  1320   46200   2356200   164934000   15173928000
| 1 1  6  90  2520  113400   7484400   681080400   81729648000
| 1 1  7 126  4284  235620  19085220  2137544640  316356606720
| 1 1  8 168  6720  436800  41932800  5577062400  981562982400
| 1 1  9 216  9936  745200  82717200 12738448800 2598643555200
| 1 1 10 270 14040 1193400 150368400 26314470000 6104957040000

			a(3) = polygorial(3,3) = A006472(3) = product of the first 3 triangular numbers = 1*3*6 = 18.
a(4) = polygorial(4,4) = A001044(4) = product of the first 4 squares = 1*4*9*16 = 576.
a(5) = polygorial(5,5) = A084939(5) = product of the first 5 pentagonal numbers = 1*5*12*22*35 = 46200.

Nathaniel Johnston, Table of n, a(n) for n = 3..100
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A084944, A085356.

A133401 := proc(n) return mul((n/2-1)*m^2-(n/2-2)*m,m=1..n): end: seq(A133401(n),n=3..15); # Nathaniel Johnston, May 05 2011

Table[Product[m*(4 - n + m*(n-2))/2, {m, 1, n}],{n, 3, 20}] (* Vaclav Kotesovec, Feb 20 2015 *)
Table[FullSimplify[(n-2)^n * Gamma[n+1] * Gamma[n+2/(n-2)] / (2^n*Gamma[2/(n-2)])],{n,3,15}] (* Vaclav Kotesovec, Feb 20 2015 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k - 2), n]]; Array[ polygorial[#, #] &, 13, 3] (* Robert G. Wilson v, Dec 13 2016 *)

a(n) ~ Pi * n^(3*n-1) / (2^(n-2) * exp(2*n+2)). - Vaclav Kotesovec, Feb 20 2015

Edited by Nathaniel Johnston, May 05 2011

A157018 Triangle T(n,k) read by rows: number of k-lists (ordered k-sets) of disjoint 2-subsets of an n-set, n>1, 0

Original entry on oeis.org

1, 3, 6, 6, 10, 30, 15, 90, 90, 21, 210, 630, 28, 420, 2520, 2520, 36, 756, 7560, 22680, 45, 1260, 18900, 113400, 113400, 55, 1980, 41580, 415800, 1247400, 66, 2970, 83160, 1247400, 7484400, 7484400, 78, 4290, 154440, 3243240, 32432400, 97297200

Offset: 2

Allan L. Edmonds and Vladeta Jovovic, Feb 21 2009

T(n,k) is also the number of involutions (unary operators) on S_n, i.e., endomorphisms U with 2k non-invariant elements such that U^2 is the identity mapping. The extension to n=1 is a(1)=0. - Stanislav Sykora, Nov 03 2016

			For n = 4 we have 12 lists: 6 1-lists: [{1,2}], [{1,3}], [{1,4}], [{2,3}], [{2,4}], [{3,4}] and 6 2-lists: [{1,2},{3,4}], [{3,4},{1,2}], [{1,3},{2,4}], [{2,4},{1,3}], [{1,4},{2,3}] and [{2,3},{1,4}].

Stanislav Sykora, Table of n, a(n) for n = 2..2501
Wikipedia, Involution (mathematics).

Cf. A000262, A000680, A087214, A100861, A126725 (row sums), A129684.

Table[n!/(2^k (n - 2 k)!), {n, 2, 13}, {k, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 04 2016 *)

nmax=100;a=vector(floor(nmax^2/4));idx=0;
for(n=2,nmax,for(k=1,n\2,a[idx++]=n!/(2^k*(n-2*k)!)));
a \\ Stanislav Sykora, Nov 03 2016

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

T(n,k) = Product_{m=1..floor(n/2)} binomial(n-2*m,2) = n!/(2^k*(n-2*k)!).

A177284 Number of permutations of 2 copies of 1..n with all adjacent differences <= 3 in absolute value.

Original entry on oeis.org

1, 1, 6, 90, 2520, 41580, 516180, 6068622, 76331906, 958679970, 11679900408, 138047313960, 1610654864328, 18649754961744, 214256589488616, 2439692058769566, 27562317214408488, 309367132582535226, 3453299423388485028, 38354816922190327314, 424048220090513056908

Offset: 0

R. H. Hardin, May 06 2010

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000680, A177283.

a(n) = (2n)!/2^n for n<=4.

a(0), a(13) from Alois P. Heinz, Jan 14 2016

Terms a(14) and beyond from Andrew Howroyd, May 14 2020

A210280 (7n)!/7^n.

Original entry on oeis.org

1, 720, 1779148800, 148953184174080000, 126983900296423931904000000, 614812159599342234168301977600000000, 11942354952042770431904585727413846016000000000

Offset: 0

Mohammad K. Azarian, Apr 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Cf. A210279, A210278, A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101

[Factorial(7*n)/7^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(7 n)!/7^n, {n, 0, 10}] (* Vincenzo Librandi, Feb 15 2013 *)

E.g.f.: 1/(1-x^7/7).

A357297 T(m,n) is the number of linear extensions of n fork-join DAGs of width m, read by downward antidiagonals.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 90, 20, 2, 1, 2520, 1680, 280, 6, 1, 113400, 369600, 277200, 9072, 24, 1, 7484400, 168168000, 1009008000, 163459296, 532224, 120, 1, 681080400, 137225088000, 9777287520000, 15205637551104, 237124952064, 49420800, 720, 1, 81729648000, 182509367040000, 207786914375040000, 4847253138540933120, 765985681152147456, 689598074880000, 6671808000, 5040, 1

Offset: 0

José E. Solsona, Feb 22 2023

The fork-join structure is a modeling structure, commonly seen for example in parallel computing, usually represented as a DAG (or poset). It has an initial "fork" vertex that spawns a number of m independent children vertices (the width) whose output edges are connected to a final "join" vertex. More generally, we can have a number n of these DAGs, each one with m+2 vertices.
The family of fork-join DAGs we are considering here can be depicted as follows (we omit the first column for n=0 because the graph is empty in this case):
m\n|    1    |      2        |          3
---------------------------------------------------
 0 |    o    |     o  o      |       o  o  o
   |    |    |     |  |      |       |  |  |
   |    o    |     o  o      |       o  o  o
---------------------------------------------------
   |    o    |     o  o      |       o  o  o
   |    |    |     |  |      |       |  |  |
 1 |    o    |     o  o      |       o  o  o
   |    |    |     |  |      |       |  |  |
   |    o    |     o  o      |       o  o  o
---------------------------------------------------
   |    o    |    o     o    |    o     o     o
   |   / \   |   / \   / \   |   / \   / \   / \
 2 |  o   o  |  o   o o   o  |  o   o o   o o   o
   |   \ /   |   \ /   \ /   |   \ /   \ /   \ /
   |    o    |    o     o    |    o     o     o
---------------------------------------------------
   |    o    |    o     o    |    o     o     o
   |   /|\   |   /|\   /|\   |   /|\   /|\   /|\
 3 |  o o o  |  o o o o o o  |  o o o o o o o o o
   |   \|/   |   \|/   \|/   |   \|/   \|/   \|/
   |    o    |    o     o    |    o     o     o
The array begins like this:
m\n|0   1         2               3                       4
-----------------------------------------------------------
 0 |1   1         6              90                    2520 ... A000680
 1 |1   1        20            1680                  369600 ... A014606
 2 |1   2       280          277200              1009008000 ... A260331
 3 |1   6      9072       163459296          15205637551104 ... A361901
 4 |1  24    532224    237124952064      765985681152147456 ... A362565
 5 |1 120  49420800 689598074880000 97981404549709824000000 ...
with columns: A000012 (n=0) and A000142 (n=1).

			T(3,1) = 6 is the number of linear extensions of one fork-join DAG of width 3. Let the DAG be labeled as follows:
     1
   / | \
  2  3  4
   \ | /
     5
Then the six linear extensions are:
  1 2 3 4 5
  1 2 4 3 5
  1 3 2 4 5
  1 3 4 1 5
  1 4 2 3 5
  1 4 3 2 5

Wikipedia, Fork-join model

Rows m = 0..4 give A000680, A014606, A260331, A361901, A362565.

Columns n = 0..1 give A000012, A000142.

(* Formula *)
T[m_, n_] := (n*(m+2))!/((m+1)^n*(m+2)^n)
(* 5 X 5 Table *)
Table[T[m, n], {m, 0, 5}, {n, 0, 5}]
(* Eight rows of the triangle *)
Table[Table[T[m, n - m], {m, 0, n}], {n, 0, 8}]
(* As a sequence *)
Flatten[Table[Table[T[m, n - m], {m, 0, n}], {n, 0, 8}]]

T(m,n) = (n*(m+2))!/((m+1)^n*(m+2)^n).

cp's OEIS Frontend

A334776 Total number of peaks in all permutations of 2 indistinguishable copies of 1..n.

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A334777 Total number of local maxima in all permutations of 2 indistinguishable copies of 1..n.

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A361651 Number T(n,k) of permutations p of [n] such that p(i), p(i+k), p(i+2k),... form an up-down sequence for i in [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A089975 Array read by ascending antidiagonals: T(n,k) is the number of n-letter words from a k-letter alphabet such that no letter appears more than twice.

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A117414 An Euler triangle.

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A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.

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A157018 Triangle T(n,k) read by rows: number of k-lists (ordered k-sets) of disjoint 2-subsets of an n-set, n>1, 0

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A177284 Number of permutations of 2 copies of 1..n with all adjacent differences <= 3 in absolute value.

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