A003121 -id:A003121 - cp's OEIS frontend

A005118 Number of simple allowable sequences on 1..n containing the permutation 12...n.

1, 1, 1, 2, 16, 768, 292864, 1100742656, 48608795688960, 29258366996258488320, 273035280663535522487992320, 44261486084874072183645699204710400, 138018895500079485095943559213817088756940800

Offset: 0

N. J. A. Sloane

For n >= 2 by the hook length formula a(n) is also the number of Young tableaux of size 1+2+...+(n-1) = n*(n-1)/2 that correspond to the partition (1,2,...n-1), i.e., triangular Young tableaux. For example, for n=5 the shape of the tableau is xxxx / xxx / xx / x. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001
Also, a(n) is the degree of the symplectic Grassmannian, the projective variety of all maximal isotropic subspaces in a complex vector space of dimension 2n-2 with a symplectic form. See Hiller's paper. - Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002
Also, for n >= 2, a(n) is the number of maximal chains in the poset of Dyck paths ordered by inclusion. - Jennifer Woodcock (Jennifer.Woodcock(AT)ugdsb.on.ca), May 21 2008
a(n) is the number of minimal decompositions of the "flip" permutation n(n-1)..21 in terms of the n-1 standard Coxeter generators (i i+1) ("reduced decompositions", cf. Stanley). As such, it is also the number of positive n-strand braid words representing the Garside braid Delta(n) (the half-turn) (cf. Epstein's book, lemma 9.1.14). - Maxime Bourrigan, Apr 04 2011
For n >= 1, the normalized volume of the subpolytope of the Birkhoff polytope obtained by taking the convex hull of all (2n)x(2n) permutation matrices corresponding to alternating permutations that also avoid the pattern 123. - Robert Davis, Dec 04 2016

D. B. A. Epstein with J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson and W. P. Thurston, Word Processing in Groups, Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..40
Omer Angel, Alexander E. Holroyd, Dan Romik, and Balint Virag, Random Sorting Networks, arXiv preprint arXiv:0609538 [math.PR], 2006.
Joerg Arndt, The a(4)=16 Young tableaux of shape [3, 2, 1].
Sara C. Billey and Peter R. W. McNamara, The contributions of Stanley to the fabric of symmetric and quasisymmetric functions, arXiv preprint, 2015.
Tobias Boege, Alessio D'Alì, Thomas Kahle, Bernd Sturmfels, The Geometry of Gaussoids, arXiv:1710.07175 [math.CO], 2017.
R. Davis and B. Sagan, Pattern-Avoiding Polytopes, 2016
FindStat - Combinatorial Statistic Finder, The number of ways to write a permutation as a minimal length product of simple transpositions
M. J. Hay, J. Schiff, and N. J. Fisch, Maximal energy extraction under discrete diffusive exchange, arXiv preprint arXiv:1508.03499 [physics.plasm-ph], 2015.
H. Hiller, Combinatorics and intersection of Schubert varieties, Comment. Math. Helv. 57 (1982), 41-59.
D. Kim, Finding k Shortest Paths in Cayley Graphs of Finite Groups, Graphs and Combinatorics 40, 120 (2024). See Formula at p. 13.
G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87. [Annotated scanned copy]
Joshua Maglione and Christopher Voll, Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration, arXiv:2410.08075 [math.CO], 2024. See pp. 34, 39.
R. P. Stanley, A combinatorial miscellany
R. P. Stanley, Ordering events in Minkowski space, arXiv:math/0501256 [math.CO], 2005.
R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5 (1984), 359-372.

Cf. A003121, A018241, A057863, A246865, A289778.

A005118 := proc(n) local i; binomial(n,2)!/product( (2*i+1)^(n-i-1), i=0..n-2 ); end;

Table[Binomial[n, 2]!/Product[(2*i + 1)^(n - i - 1), {i, 0, n - 2}], {n, 0, 10}] (* T. D. Noe, May 29 2012 *)

a(n) = C(n, 2)!/(1^{n-1} * 3^{n-2} *...* (2n-3)^1 ).
a(n) = (n*(n-1)/2)!/A057863(n-1) (n>=1). - Emeric Deutsch, May 21 2004
a(n) = A153452(A002110(n-1)). - Naohiro Nomoto, Jan 01 2009
From Alois P. Heinz, Nov 18 2012: (Start)
a(n+1) = A219272(A000217(n),n) = A219274(A000217(n),n) = A219311(A000217(n),n).
a(n) = A193536(n,A000217(n-1)) = A193629(n,A000217(n-1)). (End)
a(n) ~ sqrt(Pi) * n^(n^2/2-n/2+23/24) * exp(n^2/4-n/2+7/24) / (A^(1/2) * 2^(n^2-n/2-7/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

Citation corrected by Matthew J. Samuel, Feb 01 2011

A282698 Irregular triangle read by rows: row n gives numbers of maximal chains of lengths n-1, n, n+1, ... in the Tamari lattice T_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 2, 2, 1, 10, 22, 22, 18, 13, 12, 1, 20, 112, 232, 382, 348, 456, 390, 420, 334, 286, 1, 35, 392, 1744, 4474, 8435, 12732, 17337, 21158, 27853, 33940, 41230, 45048, 50752, 41826, 33592, 1, 56, 1092, 9220, 40414, 123704, 276324, 550932, 917884

Offset: 1

N. J. A. Sloane, Feb 25 2017

Nelson (2017) gives first nine columns of the transposed triangle.

			Triangle begins:
  1;
  1;
  1,  1;
  1,  4,   2,   2;
  1, 10,  22,  22,  18,  13,  12;
  1, 20, 112, 232, 382, 348, 456, 390, 420, 334, 286;
  ...
The transposed triangle, as given by Nelson, begins:
  1, 1, 1, 1,  1,   1,     1,        1,          1, ...
        1, 4, 10,  20,    35,       56,         84, ...
           2, 22, 112,   392,     1092,       2604, ...
           2, 22, 232,  1744,     9220,      37444, ...
              18, 382,  4474,    40414,     280214, ...
              13, 348,  8435,   123704,    1321879, ...
              12, 456, 12732,   276324,    4578596, ...
                  390, 17337,   550932,   12512827, ...
                  420, 21158,   917884,   29499764, ...
                  334, 27853,  1510834,   62132126, ...
                  286, 33940,  2166460,  120837274, ...
                       41230,  3370312,  221484557, ...
                       45048,  4810150,  393364848, ...
                       50752,  7264302,  666955139, ...
                       41826, 10435954, 1134705692, ...
                       33592, 15227802, 1933708535, ...
  ...

Alois P. Heinz, Rows n = 1..14, flattened
Luke Nelson, A recursion on maximal chains in the Tamari lattices, Discrete Mathematics 340.4 (2017): 661-677.
Luke Nelson, A recursion on maximal chains in the Tamari lattices, arXiv:1709.02987 [math.CO], (2017)

Row sums give A027686.

Right border gives A003121(n-1).

s:= proc(n) s(n):=`if`(n=0, [], [s(n-1), []]) end:
f:= l-> l=[] or l[1]=[] and f(l[2]):
v:= proc(l) v(l):=`if`(f(l), [], [`if`(l[1]<>[],
      [l[1][1], [l[1][2], l[2]]], [][]),
      seq([w, l[2]], w=v(l[1])), seq([l[1], w], w=v(l[2]))])
    end:
p:= proc(l) p(l):=`if`(f(l), 1, add(expand(x*p(w)), w=v(l))) end:
T:= n-> (h-> seq(coeff(h, x, i), i=ldegree(h)..degree(h)))(p(s(n))):
seq(T(n), n=1..8);  # Alois P. Heinz, Jan 02 2018

More terms from Alois P. Heinz, Jan 02 2018

A007724 Even minus odd extensions of truncated 3 X 2n grid diagram.

Original entry on oeis.org

2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100

Offset: 2

Frank Ruskey

Number of standard tableaux of shapes (n-1,n-1,k), k=0,1,...,n-1. Example: a(3)=12 because there are 2, 5 and 5 standard tableaux of shapes (2,2), (2,2,1) and (2,2,2), respectively. - Emeric Deutsch, May 25 2004
From Joel B. Lewis, Oct 05 2009: (Start)
Also the number of standard shifted Young tableaux of shape (n+1, n, n-1).
Also the number of 2143-avoiding up-down permutations of length 2n - 1. (End)

J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.
F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77-101.
Dennis White, Sign-balanced posets

Cf. A003121.
2143-avoiding up-down permutations of length 2n are given by A005789. - Joel B. Lewis, Oct 05 2009
After corrections, is very similar to A217800.

A007724 := proc(n)
    combinat[multinomial](3*n,n-1,n,n+1)/n/(2*n-1)/(2*n+1) ;
end proc:
seq(A007724(n),n=2..40) ; # R. J. Mathar, Jul 07 2023

Table[(3*n)!/((n-1)!*n!*(n+1)!)/(n*(2*n-1)*(2*n+1)),{n,2,10}] (* Vaclav Kotesovec, Nov 13 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 19}] (* Michael De Vlieger, Aug 22 2016 *)

{a(n) = if(n<2, 0, (3*n)!/((2*n+1) * (2*n-1) * (n+1)! * n!^2))}; /* Michael Somos, Jul 04 2020 */

a(n) = multinomial(3n; n-1, n, n+1)/(n(2n-1)(2n+1)).
a(n) ~ 3^(3*n+1/2) / (8*Pi*n^4). - Vaclav Kotesovec, Nov 13 2014
D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -3*(3*n-1)*(2*n-3)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Jul 07 2023

More terms from James Sellers, Dec 24 1999

a(16)-a(18) corrected and a(19)-a(20) added by Alois P. Heinz, Aug 22 2016

A004065 Define predecessors of n, P(n), to consist of numbers whose binary representation is obtained from that of n by replacing 10 with 01 or changing a final 1 to a 0; then a(0)=1, a(n) = Sum a(P(n)), n>0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 7, 5, 12, 12, 12, 1, 4, 9, 16, 14, 42, 54, 66, 14, 56, 110, 176, 110, 286, 286, 286, 1, 5, 14, 30, 28, 100, 154, 220, 42, 198, 462, 858, 572, 1716, 2002, 2288, 42, 240, 702, 1560, 1274, 4550, 6552, 8840, 1274, 5824, 12376

Offset: 0

David W. Wilson, Jan 29 2000

			E.g. 201 = 11001001, so P(201) = {169, 197, 200}, a(201) = a(169) + a(197) + a(200).

Alois P. Heinz, Table of n, a(n) for n = 0..8191

Cf. A003121.

P:= proc(n) local h, i, m, s, t;
      t:= irem(n, 2, 'm');
      s:= `if`(t=1, {n-1}, {});
      for i from 0 while m>0 do h:= irem(m, 2, 'm');
        if h=1 and t=0 then s:= s union {n- 2^i} fi; t:=h
      od; s
    end:
a:= proc(n) a(n):= `if`(n=0, 1, add(a(k), k=P(n))) end:
seq (a(n), n=0..80);  # Alois P. Heinz, Jul 06 2012

P[n_] := Module[{h, i, m, s, t}, {m, t} = QuotientRemainder[n, 2]; s = If[t == 1, {n-1}, {}]; For[i = 0, m>0, i++, {m, h} = QuotientRemainder[m, 2]; If[h == 1 && t == 0, s = s ~Union~ {n-2^i}]; t = h]; s]; a[n_] := a[n] = If[n == 0, 1, Sum[a[k], {k, P[n]}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 11 2015, after Alois P. Heinz *)

a(2^n-1) = A003121(n).

Entry revised by N. J. A. Sloane, Jun 14 2012

A018241 Number of simple allowable sequences on 1..n.

Original entry on oeis.org

1, 1, 2, 32, 4608, 7028736, 132089118720, 34998332896051200, 147462169661142781132800, 11008782516353752266715850342400, 16061608070479103314001351327405309952000, 500842967990688435516159987675099451681186775040000

Offset: 1

N. J. A. Sloane

J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87.

Alois P. Heinz, Table of n, a(n) for n = 1..40
G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87. [Annotated scanned copy]
R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5 (1984), 359-372.

Cf. A003121, A005118, A074962.

A018241 := proc(n) local i; (n-2)!*binomial(n,2)!/product( (2*i+1)^(n-i-1), i=0..n-2 ); end;

a[n_] := (n*(n-1)/2)!*(n-2)! / Product[ (2i+1)^(n-i-1), {i, 0, n-2}]; a[1] = 1; Table[ a[n], {n, 1, 11}] (* Jean-François Alcover, Jan 25 2012 *)

a(n) = (n-2)!*C(n,2)! / (1^{n-1} * 3^{n-2} * ... * (2n-3)^1).

a(n) ~ Pi * exp(n^2/4 - 3*n/2 + 7/24) * n^(n^2/2 + n/2 - 13/24) / (A^(1/2) * 2^(n^2 - n/2 - 19/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

A061343 Number of standard shifted tableaux with n entries.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 27, 63, 154, 398, 1055, 2970, 8503, 25651, 78483, 250487, 811802, 2723130, 9295483, 32653552, 116866283, 428464743, 1600474365, 6102119282, 23690388631, 93631999867, 376561553417, 1538997717423, 6395852269479, 26978392034357, 115628083386280, 502520979828775

Offset: 1

V. Reiner and D. White (reiner(AT)math.umn.edu), Jun 07 2001

Number of ballot sequences (see A000085) where the number of occurrences of k in any prefix is strictly greater than the number of occurrences of k+1. - Joerg Arndt, May 21 2016

			From _Joerg Arndt_, May 21 2016: (Start)
The a(7) = 27 tableaux correspond to the following ballot sequences (dots denote zeros).
##:     ballot sequence          partition
01:    [ . . . . . . . ]       [ 7 . . . . . . ]
02:    [ . . . . . . 1 ]       [ 6 1 . . . . . ]
03:    [ . . . . . 1 . ]       [ 6 1 . . . . . ]
04:    [ . . . . . 1 1 ]       [ 5 2 . . . . . ]
05:    [ . . . . 1 . . ]       [ 6 1 . . . . . ]
06:    [ . . . . 1 . 1 ]       [ 5 2 . . . . . ]
07:    [ . . . . 1 1 . ]       [ 5 2 . . . . . ]
08:    [ . . . . 1 1 1 ]       [ 4 3 . . . . . ]
09:    [ . . . . 1 1 2 ]       [ 4 2 1 . . . . ]
10:    [ . . . 1 . . . ]       [ 6 1 . . . . . ]
11:    [ . . . 1 . . 1 ]       [ 5 2 . . . . . ]
12:    [ . . . 1 . 1 . ]       [ 5 2 . . . . . ]
13:    [ . . . 1 . 1 1 ]       [ 4 3 . . . . . ]
14:    [ . . . 1 . 1 2 ]       [ 4 2 1 . . . . ]
15:    [ . . . 1 1 . . ]       [ 5 2 . . . . . ]
16:    [ . . . 1 1 . 1 ]       [ 4 3 . . . . . ]
17:    [ . . . 1 1 . 2 ]       [ 4 2 1 . . . . ]
18:    [ . . . 1 1 2 . ]       [ 4 2 1 . . . . ]
19:    [ . . 1 . . . . ]       [ 6 1 . . . . . ]
20:    [ . . 1 . . . 1 ]       [ 5 2 . . . . . ]
21:    [ . . 1 . . 1 . ]       [ 5 2 . . . . . ]
22:    [ . . 1 . . 1 1 ]       [ 4 3 . . . . . ]
23:    [ . . 1 . . 1 2 ]       [ 4 2 1 . . . . ]
24:    [ . . 1 . 1 . . ]       [ 5 2 . . . . . ]
25:    [ . . 1 . 1 . 1 ]       [ 4 3 . . . . . ]
26:    [ . . 1 . 1 . 2 ]       [ 4 2 1 . . . . ]
27:    [ . . 1 . 1 2 . ]       [ 4 2 1 . . . . ]
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 3 (Sorting and searching), page 71, Section 5.1.4, Exercise 21 (page 67 in the second edition).

Joerg Arndt, Table of n, a(n) for n = 1..101
Joerg Arndt, PARI/GP script to compute terms.
R. Srinivasan, On a theorem of Thrall in combinatorial analysis, The American Mathematical Monthly, 70(1), 1963, pp. 41-44.
R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.

Cf. A000085, A003121 (strict ballot sequences with partition [j, j-1, ..., 3, 2, 1]).

a(n) is the sum over all partitions into distinct parts of Thrall's formula (4) on page 83, see the PARI script arndt-A061343.gp. - Joerg Arndt, May 09 2013

More terms from Joerg Arndt, May 08 2013

A213457 Intertwining numbers.

Original entry on oeis.org

1, 1, 2, 10, 148, 7384, 1380960, 1058349286

Offset: 1

N. J. A. Sloane

a(4)=10 for example is the number of ways of arranging 1 a, 2 b's, 3 c's and 4 d's so that if we look at any two letters, i and j say, with i

If the condition is imposed only on pairs of consecutive letters, we get A003121.

			The 10 sequences for n=4 are dcbadcbdcd dcbadcdbcd dcbdacbdcd dcbdacdbcd dcbdcabdcd dcbdcadbcd dcbdcdabcd dcdbacdbcd dcdbcadbcd dcdbcdabcd.
For example in dcbdacdbcd we see
..ba...b..
.cb..c.bc.
d.bd..db.d
dc.d.cd.cd

C. L. Mallows, Letter to N. J. A. Sloane, Nov 11 1980
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Perl program

Cf. A003121.

Entry revised and given a new A-number by N. J. A. Sloane, Jun 13 2012. The old entry was A004065
a(7) from David W. Wilson, Dec 11 1999
Definition clarified by David Applegate, Jun 14 2012
a(8) from Georg Fischer, Mar 21 2018

A237749 The number of possible orderings of the real numbers xi*xj (i <= j), subject to the constraint that x1 > x2 > ... > xn > 0.

Original entry on oeis.org

1, 1, 1, 2, 10, 114, 2608, 107498, 7325650, 771505180

Offset: 0

Nathaniel Johnston, Feb 12 2014

Also, the number of possible orderings of sums xi + xj (i <= j), subject to the constraint that x1 > x2 > ... > xn. Cf. A231085.
Also, the minimum number of linear matrix inequalities needed to characterize the eigenvalues of quantum states that are "PPT from spectrum" when the local dimension is n (see Hildebrand link).
A003121 is an upper bound on this sequence.
Alternately, the number of combinatorially different Golomb rulers with n markings (Beck-Bogart-Pham). - Charles R Greathouse IV, Feb 18 2014
From Jon E. Schoenfield, Jul 03 2015: (Start)
The terms of this sequence would remain unchanged if it were required that each value of xi (and hence each pairwise product xi*xj) be an integer, and the addition of such a constraint suggests a systematic (albeit impractical for larger values of n) way to search through sets of n values of x to find a set that yields each of the a(n) possible orderings of the pairwise products: for x1 = n, n+1, n+2, ..., test every combination of n distinct positive integers of which x1 is the largest. Let M(n) be the smallest integer such that each of the a(n) possible orderings results from at least one combination of integers x1, x2, ..., xn where M(n) >= x1 > x2 > ... > xn; then values of M(n) for n = 2..6 are 2, 5, 13, 29, and 68, respectively.
For any given value of x1, the number of distinct orderings of pairwise products resulting from the binomial(x1-1, n-1) possible combinations of the remaining integers x2..xn provides a lower bound L(x1) for a(n). In general, L(x1) is not monotonically nondecreasing; e.g., for n=6, the (weak) lower bound on a(6)=2608 provided by L(33) is 2428, and L(34)=2423 is weaker still. However -- at least up through n=6 -- each of the a(n) possible orderings results from at least one combination where x1 is exactly M(n); e.g., at n=6, one of the 2608 orderings is missing among all binomial(67,6) = 99,795,696 combinations where x1 < 68, but all 2608 are present among the binomial(67,5) = 9,657,648 combinations where x1=68.
For all n up to at least 6, the number of orderings found among all combinations where x1 < M(n) is a(n)-1, and the one missing ordering of the pairwise products is the one in which xj*xn > (x(j+1))^2 for j=1..n-1. (End)
Number of paths towards synchronization in the transition diagram associated with the Laplacian system over the complete graph K_N, corresponding to ordered initial conditions x_1 < x_2 < ... < x_N. - Andrea Arlette España, Nov 14 2022

			a(3) = 2 because there are 2 possible orderings of the 6 products a1^2, a2^2, a3^2, a1*a2, a1*a3, a2*a3. Specifically, these orderings are:
a1^2 > a1a2 > a2^2 > a1a3 > a2a3 > a3^2 and
a1^2 > a1a2 > a1a3 > a2^2 > a2a3 > a3^2.

S. Arunachalam, N. Johnston, and V. Russo, Is separability from spectrum determined by the partial transpose?, arXiv preprint arXiv:1405.5853 [quant-ph], 2014-2015.
Matthias Beck, Tristram Bogart, and Tu Pham, Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs, arXiv:1110.6154 [math.CO], 2011, Section 5.
A. España, X. Leoncini, and E. Ugalde, Combinatorics of the paths towards synchronization, arXiv:2205.05948 [math.DS], 2022.
R. Hildebrand, Positive partial transpose from spectra. Phys. Rev. A, 76 (5) (2007) 052325, [arXiv], arXiv:quant-ph/0502170, 2005.
Nathaniel Johnston, Counting the possible orderings of pairwise multiplication
Nathaniel Johnston and Olivia MacLean, Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States, arXiv:1807.06897 [quant-ph], 2018.
Antti Laaksonen, Counting Orderings of Sums
Steven J. Miller and Carsten Peterson, A geometric perspective on the MSTD question, arXiv:1709.00606 [math.CO], 2017.
Tu Pham, Enumeration of Golomb Rulers, Master's Thesis (2011) Table 3.1.

Cf. A003121, A231074, A231085, A259762.

a(7) copied from Tu Pham by Charles R Greathouse IV, Feb 18 2014

a(8)-a(9) from Antti Laaksonen, Jan 10 2019

A376162 Number of ordered partitions of S={(i,j):1 <= i , j <= n} where for every i and j the pairs (i+1,j) and (i,j+1) are in a later part than the part containing the pair (i,j), and the pairs (i,j), (j,i) are in the same part.

Original entry on oeis.org

1, 1, 3, 39, 2905, 1538369, 6904262355, 304662492057063, 150347237334006997801, 929721796071361437087789041, 79773595676787229793797978773561927, 104165556509336140832819242491033872033130063, 2252283824141388832759484222915451435885285752729087857

Offset: 1

Kevin O'Bryant, Sep 12 2024

nonn

Ordered partitions are also called weak orderings.

Any such ordered partition can be written as a list of pairs (i,j) with 1 <= i <= j <= n, with either "=" or "<" between each pair, and so that (i,j) appears in the list before (i+1,j) (if i

Given any set A={a_1<...

Equivalently, a(n) is the number of n X n symmetric matrices whose values cover an initial interval of positive integers and whose rows have values which are strictly increasing. - Andrew Howroyd, Sep 15 2024

			For n=2 the a(2)=1 ordered partition is {(1,1)}<{(2,1),(1,2)}<{(2,2)}. We can encode this as 11<12<22, writing "ij" for the pair (i,j).
For n=3 one of the a(3)=3 ordered partitions is {(1,1)}<{(1,2),(2,1)}<{(1,3),(3,1),(2,2)}<{(2,3),(3,2)}<{(3,3)}, which is encoded as either 11<12<13=22<23<33 or 11<12<22=13<23<33. The other two ordered partitions can be encoded as 11<12<22<13<23<33 and 11<12<13<22<23<33.
From _Andrew Howroyd_, Sep 15 2024: (Start)
The a(3) = 3 symmetric matrices are:
    [1 2 3]   [1 2 3]   [1 2 4]
    [2 3 4]   [2 4 5]   [2 3 5]
    [3 4 5]   [3 5 6]   [4 5 6]
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..20

Cf. A000670, A006541, A003121, A374514.

\\ See PARI link in A374514 for program code.
vector(10, n, A376162(n)) \\ Andrew Howroyd, Sep 16 2024

a(n) <= A000670(n*(n+1)/2).

a(7) onwards from Andrew Howroyd, Sep 15 2024

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cp's OEIS Frontend

A083568 Duplicate of A003121.

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A005118 Number of simple allowable sequences on 1..n containing the permutation 12...n.

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A282698 Irregular triangle read by rows: row n gives numbers of maximal chains of lengths n-1, n, n+1, ... in the Tamari lattice T_n.

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A007724 Even minus odd extensions of truncated 3 X 2n grid diagram.

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A004065 Define predecessors of n, P(n), to consist of numbers whose binary representation is obtained from that of n by replacing 10 with 01 or changing a final 1 to a 0; then a(0)=1, a(n) = Sum a(P(n)), n>0.

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A018241 Number of simple allowable sequences on 1..n.

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A061343 Number of standard shifted tableaux with n entries.

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A213457 Intertwining numbers.

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