A005130 -id:A005130 - cp's OEIS frontend

A005162 Number of alternating sign n X n matrices symmetric with respect to both diagonals.

1, 2, 3, 8, 15, 52, 126, 568, 1782, 10436, 42471, 323144, 1706562, 16866856, 115640460, 1484714416, 13216815036, 220426128584, 2548124192970

Offset: 1

N. J. A. Sloane and Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

Mireille Bousquet-Mélou and Laurent Habsieger, Sur les matrices à signes alternants, [On alternating-sign matrices] in Formal power series and algebraic combinatorics (Montreal, PQ, 1992) pp. 19-32; Discrete Math. 139 (1995), 57-72. See Table 1, p. 71.
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Robbins gives a simple (conjectured) formula.

A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.

Original entry on oeis.org

0, 0, 1, 2, 5, 7, 11, 15, 21, 25, 32, 38, 47

Offset: 1

N. J. A. Sloane

The known values a = a(n) and upper bounds U (usually A032765(n)) with name of discoverer of the arrangement when known are as follows:
   n       a    U  [Found by]
------------------------------
     0    0
     0    0
     1    1
     2    2
     5    5
     7    7
    11   11
    15   16
    21   21
    25   25  [Grünbaum]
    32   33  [32-triangle solutions found by Honma and Kabanovitch; proved maximal by Savchuk 2025]
    38   38  [Kabanovitch]
    47   47  [Kabanovitch]
 >= 53   54  [Bader]
    65   65  [Suzuki]
    72   72  [Bader]
    85   85  [Bader]
 >= 93   94  [Bader]
   107  107  [Wood]
>= 116  117  [Wood]
   133  133  [Savchuk]
>= 143  144  [Savchuk]
   161  161  [Savchuk]
>= 172  173  [Savchuk]
   191  191  [Bartholdi]
     ?  205
   225  225  [Savchuk]
     ?  239
   261  261  [Bartholdi]
     ?  276
   299  299  [Wood]
     ?  316
   341  341  [Bartholdi]
Ed Pegg's web page gives the upper bound for a(6) as 8. But by considering all possible arrangements of 6 lines - the sixth term of A048872 - one can see that 8 is impossible. - N. J. A. Sloane, Nov 11 2007
Although they are somewhat similar, this sequence is strictly different from A084935, since A084935(12) = 48 exceeds the upper bound on a(12) from A032765. - Floor van Lamoen, Nov 16 2005
The name is sometimes incorrectly entered as "Kodon" triangles.
Named after the Japanese puzzle expert and mathematics teacher Kobon Fujimura (1903-1983). - Amiram Eldar, Jun 19 2021

			a(17) = 85 because the a configuration with 85 exists meeting the upper bound.

Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, "On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles", 2008, in Goodman, Jacob E.; Pach, János; Pollack, Richard (eds.), Surveys on Discrete and Computational Geometry: Proceedings of the 3rd AMS-IMS-SIAM Joint Summer Research Conference "Discrete and Computational Geometry—Twenty Years Later" held in Snowbird, UT, June 18-22, 2006, Contemporary Mathematics, vol. 453, Providence, Rhode Island: American Mathematical Society, pp. 105-116, doi:10.1090/conm/453/08797, ISBN 978-0-8218-4239-3, MR 2405679
Martin Gardner, Wheels, Life and Other Mathematical Amusements, Freeman, NY, 1983, pp. 170, 171, 178. Mentions that the problem was invented by Kobon Fujimura.
Branko Grünbaum, Convex Polytopes, Wiley, NY, 1967; p. 400 shows that a(10) >= 25.
Viatcheslav Kabanovitch, Kobon Triangle Solutions, Sharada (Charade, by the Russian puzzle club Diogen), pp. 1-2, June 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Johannes Bader, Kobon Triangles.
Johannes Bader, Kobon Triangles. [Cached copy, with permission, pdf format]
Johannes Bader, Illustration showing a(17)=85, Nov 28 2007.
Johannes Bader, Illustration showing a(17)=85, Nov 28 2007. [Cached copy, with permission]
Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles, arXiv:0706.0723 [math.CO], 2007.
Nicolas Bartholdi, Jérémy Blanc, Sébastien Loisel, and Pavlo Savchuk, Illustration showing a(33) = 341, 2008.
Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007.
Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007. [Cached copy, with permission]
Martin Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
S. Honma, 三角形の最大数
S. Honma, Illustration showing a(10)>=25
S. Honma, Illustration showing a(11)>=32
S. Honma, n本の直線でn(n-2)/3個の三角形が出来る条件についての考察: part 1, part 2, part 3.
Ed Pegg, Jr., Kobon triangles, 2006.
Ed Pegg, Jr., Kobon Triangles, 2006. [Cached copy, with permission, pdf format]
Luis Felipe Prieto-Martínez, A list of problems in Plane Geometry with simple statement that remain unsolved, arXiv:2104.09324 [math.HO], 2021.
Pavlo Savchuk, Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening, arXiv:2507.07951 [math.CO], 2025. See pp. 1, 17.
Pavlo Savchuk, Illustration showing a(21) = 133
Pavlo Savchuk, Illustration showing a(23) = 161
Pavlo Savchuk, Corresponding pseudo-line arrangement for the optimal n=23 solution
Pavlo Savchuk, Illustration showing a(27) = 225
Pavlo Savchuk, Illustration showing a(22) >= 143
Pavlo Savchuk, Illustration showing a(24) >= 172
N. J. A. Sloane, Illustration for a(5) = 5 (a pentagram)
Alexandre Wajnberg, Illustration showing a(10) >= 25. [A different construction from Grünbaum's]
Eric Weisstein's World of Mathematics, Kobon Triangle.
Kyle Wood, Illustration showing a(19) = 107
Kyle Wood, Illustration showing a(20) >= 116
Kyle Wood, Illustration showing a(31) = 299

An upper bound on this sequence is given by A032765.
For any odd n > 1, if n == 1 (mod 6), a(n) <= (n^2 - (2n + 2))/3; in other odd cases, a(n) <= (n^2 - 2n)/3. For any even n  > 0, if n == 4 (mod 6), a(n) <= (n^2 - (2n + 2))/3, otherwise a(n) <= (n^2 - 2n)/3. - Sergey Pavlov, Feb 11 2017
The upper bound for even n can be improved: floor(n(n-7/3)/3), proven by Bartholdi et. al.

a(15) = 65 found by Toshitaka Suzuki on Oct 02 2005. - Eric W. Weisstein, Oct 04 2005
Grünbaum reference from Anthony Labarre, Dec 19 2005
Additional links to Japanese web sites from Alexandre Wajnberg, Dec 29 2005 and Anthony Labarre, Dec 30 2005
A perfect solution for 13 lines was found in 1999 by Kabanovitch. - Ed Pegg Jr, Feb 08 2006
Updated with results from Johannes Bader (johannes.bader(AT)tik.ee.ethz.ch), Dec 06 2007, who says "Acknowledgments and dedication to Corinne Thomet".
a(11)-a(13) from Eric W. Weisstein, Jul 26 2025

A029729 Degree of the variety of pairs of commuting n X n matrices.

Original entry on oeis.org

1, 3, 31, 1145, 154881, 77899563, 147226330175, 1053765855157617, 28736455088578690945, 3000127124463666294963283, 1203831304687539089648950490463, 1862632561783036151478238040096092649, 11143500837236042423379349834982088594105985

Offset: 1

Nolan R. Wallach (nwallach(AT)euclid.ucsd.edu), Dec 11 1999

Also, ratio of vector elements of the ground state in the loop representation of the braid-monoid Hamiltonian H = Sum_i (3 - 2 e_i - b_i) with size 2n and periodic boundary conditions. Specifically the smallest element that corresponds to a non-crossing chord diagram, divided by the overall smallest element. We reduce the standard braid-monoid algebra to the Brauer algebra B_{2n}(1). - B. Nienhuis & J. de Gier (B.Nienhuis(AT)UvA.NL), May 13 2004. For a proof that this is the same sequence, see the articles by P. Di Francesco and P. Zinn-Justin and A. Knutson and P. Zinn-Justin.
These numbers arise in a similar way to A005130 and related sequences appear in the groundstate of the integrable Temperley-Lieb Hamiltonian.
It is also the weighted enumeration of lattice paths on an n X n square lattice going from the left side to the top side, with same initial and final orders of paths, and with a weight of 2 per vertex where a path turns 90 degrees. - Paul Zinn-Justin, Mar 05 2023

			n=1: Degree of C X C which is 1. n=2: The degree can be calculated by hand to be 3. n=3: See Macaulay manual (link above): one of steps in proof that variety for 3 X 3 is Cohen-Macaulay is to compute the degree which is 31. (n=4) Matt Clegg (CS at UCSD) and Nolan Wallach using 10 Sun Workstations and a distributed Grobner Basis package (in 1993).
(2(e1 + e2 + e3 + e4) + b1 + b2 + b3 + b4)(G + G e2 + b2)(e1 e3 b2) = 12 (G + G e2 + b2)(e1 e3 b2) with G = 3, therefore a(2) = 3

Paul Zinn-Justin, Table of n, a(n) for n = 1..16
Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
P. Di Francesco and P. Zinn-Justin, Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, Comm. Math. Phys., 262(2):459-487, 2006; arXiv preprint, arXiv:math-ph/0412031, 2004-2005.
A. Garbali and P. Zinn-Justin, Shuffle algebras, lattice paths and the commuting scheme, arXiv:2110.07155 [math.RT], 2021-2022. See also Macaulay2 code to generate the sequence.
A. Knutson and P. Zinn-Justin, A scheme related to the Brauer loop model, Adv. Math., 214(1):40-77, 2007, arXiv preprint, arXiv:math/0503224 [math.AG], 2005-2006.
Macaulay 2 Manual, Test of matrix routines, Viewed May 03 2016.
M. J. Martins, B. Nienhuis, and R. Rietman, An Intersecting Loop Model as a Solvable Super Spin Chain, arXiv:cond-mat/9709051 [cond-mat.stat-mech], 1997; Phys. Rev. Lett. Vol. 81 (1998) pp. 504-507.
Ada Stelzer and Alexander Yong, Combinatorial commutative algebra rules, arXiv:2306.00737 [math.CO], 2023.

Cf. A005130.

There is a formula in terms of divided differences operators (too complicated to reproduce here).

Entry revised based on comments from Paul Zinn-Justin, Mar 14 2005

Terms a(12) and beyond from Paul Zinn-Justin, Mar 05 2023

A051055 'Connected' alternating sign n X n matrices, i.e., not made from smaller blocks.

Original entry on oeis.org

0, 1, 0, 1, 2, 59, 1092, 51412, 3420384, 382912420, 68021283668, 19474443244283, 9025228384142396, 6825775070789988992, 8486240219059861120000, 17454179683586670023001218, 59698062960218238908531091872

Offset: 0

Don Knuth

A003827 factors out the singleton components only, but many alternating sign matrices can be decomposed into larger pieces.

			a(4)=2 because of the alternating sign matrices {{0,1,0,0},{1,-1,1,0},{0,1,-1,1},{0,0,1,0}} and {{0,0,1,0},{0,1,-1,1},{1,-1,1,0},{0,1,0,0}}.

Alois P. Heinz, Table of n, a(n) for n = 0..90

Cf. A003827, A005130.

r[n_] = Product[(3k+1)!/(n+k)!, {k, 0, n-1}] ; a[n_] := a[n] = r[n] - (1/n)*Sum[k*Binomial[n, k]^2*r[n-k]*a[k], {k, 0, n-1}]; a[0] = 0; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Aug 01 2011, after Vladeta Jovovic *)

Sum_{k>=0} a(k)z^k/k!^2 = log(Sum_{k>=0} r(k)z^k/k!^2) where r(k) is the k-th Robbins number A005130(n).

a(n) = r(n) - (1/n)*Sum_{k=0..n-1} k*binomial(n, k)^2*r(n-k)*a(k), n > 0, a(0)=0, where r(k) is the k-th Robbins number A005130(n). - Vladeta Jovovic, Mar 16 2000

More terms from Vladeta Jovovic, Mar 16 2000

A059476 Number of 4n X 4n quarter-turn symmetric alternating-sign matrices (QTSASM's).

Original entry on oeis.org

1, 2, 40, 6860, 9779616, 114640611228, 10995014015567296, 8604220484134405095680, 54849585336544518476372715520, 2845241019832977820277705514596768000, 1200139922362764777157943140382976769515000000

Offset: 0

N. J. A. Sloane, Feb 04 2001

nonn

G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.

a(n) = A059475(n)*A005130(n)^2.

A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

Original entry on oeis.org

1, 1, 6, 1440, 36288000, 184354652160000, 309071606732292096000000, 254046582743105184577722777600000000, 142518177743863255019484504453155074867200000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*3!*1!*2!) = 34560/24 = 1440.

Cf. A000178, A001700, A005130, A009963, A110131, A112332, A107254, A000142.

[1] cat [(&*[Factorial(n+k)/Factorial(k+1): k in [0..n-1]]): n in [1..10]]; // G. C. Greubel, May 21 2019

Table[Product[(n+k)!/(k+1)!,{k,0,n-1}],{n,0,10}] (* Alexander Adamchuk, Jul 10 2006 *)
a[n_] := (BarnesG[1 + 2 n] n! ((BarnesG[2 + n] Gamma[2 + n])/ BarnesG[3 + n])^(-1 + n))/BarnesG[2 + n]^2; Table[a[n], {n, 0, 10}] (* Peter Luschny, May 20 2019 *)

{a(n) = prod(k=0,n-1, (n+k)!/(k+1)!)}; \\ G. C. Greubel, May 21 2019

[product(factorial(n+k)/factorial(k+1) for k in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 21 2019

a(n) = (n+1)!*(n+2)!*...*(2n-1)!/(1!*2!*...*(n-1)!).
a(n) = A000178(2n-1)/(A000178(n)*A000178(n-1)).
a(n) = A079478(n)/A001813(n).
a(n) = A079478(n-1)*A006963(n+1).
a(n) = A107251(n)/A000108(n).
a(n) = A107251(n-1)*A009445(n-1).
a(n) = A107254(n)/A000142(n).
a(n) = A009963(2n-1, n-1).
a(n) = A009963(2n-1, n).
a(n) = (G(1+2*n)*n!*((G(2+n)*Gamma(2+n))/G(3+n))^(n-1))/G(2+n)^2, where G(x) is the Barnes G function. - Peter Luschny, May 20 2019
a(n) ~ A * 2^(2*n^2 - 7/12) * n^(n^2 - n - 5/12) / (sqrt(Pi) * exp(3*n^2/2 - n + 1/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 21 2019

A231498 Dimensions of algebraic generators of Hopf algebra ASM.

Original entry on oeis.org

0, 1, 1, 4, 29, 343, 6536, 202890, 10403135, 889855638, 127697994191, 30835684607975, 12548984661501958, 8614128967701807873, 9978249783441321029726, 19509076435279656169459710, 64388117502939421121766340671, 358740327955249666636651777774959

Offset: 0

N. J. A. Sloane, Nov 09 2013

nonn

Number of alternating sign matrices of size n which are indecomposable by direct sum. - Ludovic Schwob, Feb 11 2023

Ludovic Schwob, Table of n, a(n) for n = 0..100
H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.

Cf. A005130.

G.f.: A(x) = 1 - 1/ASM(x) where ASM(x) is the g.f. of A005130. - Ludovic Schwob, Feb 11 2023

More terms from Ludovic Schwob, Feb 11 2023

A293930 Number of circularly chained n X n alternating sign matrices.

Original entry on oeis.org

1, 2, 20, 40, 3430, 6860

Offset: 1

N. J. A. Sloane, Oct 19 2017

Heuer, Dylan, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, and Corey Vorland. "Chained permutations and alternating sign matrices - Inspired by three-person chess." Discrete Mathematics 340, no. 12 (2017): 2732-2752. Also arXiv:1611.03387

Cf. A005130, A293178, A293179, A293180.

A384062 Number of maximal antichains in the Bruhat order of type A_n.

Original entry on oeis.org

2, 4, 43, 183667

Offset: 1

Dmitry I. Ignatov, May 18 2025

The number of maximal antichains in the Bruhat order of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).

			For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The maximal antichains are {1} and {s1}.
For n=2 the line (Hasse) diagram is below.
       s1*s2*s1
        /   \
      s2*s1 s1*s2
       |  X  |
       s2    s1
        \   /
          1
The set of maximal antichains is {{1}, {s2, s1}, {s2*s1, s1*s2}, {s1*s2*s1}}.

A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
V. V. Deodhar, On Bruhat ordering and weight-lattice ordering for a Weyl group, Indagationes Mathematicae, vol. 81, 1 (1978), 423-435.

Cf. A000142 (the order size), A005130 (the size of Dedekind-MacNeille completion), A384061.

A059491 Expansion of generating function A_{QT}^(1)(4n;3).

Original entry on oeis.org

1, 1, 6, 189, 30618, 25332021, 106698472452, 2283997201168644, 248218139523497121576, 136861610819571430116630660, 382684747771430768732371981946100, 5424628155237728987530088501811168904125, 389729317367139375014273384868937660572301897500

Offset: 0

N. J. A. Sloane, Feb 04 2001

G. C. Greubel, Table of n, a(n) for n = 0..50
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001 [Th. 5].

f[n_] := Product[(3 k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[3^(n*(n - 1)/2)*f[n], {n,0,20}] (* G. C. Greubel, Sep 10 2017 *)

a(n) = 3^(n*(n-1)/2)*A005130(n).

a(n+1) is the Hankel transform of A097188. Odd terms occur in a(n+1) at positions given by 2*A000975(n). - Paul Barry, Feb 09 2007

cp's OEIS Frontend

A005162 Number of alternating sign n X n matrices symmetric with respect to both diagonals.

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A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.

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A029729 Degree of the variety of pairs of commuting n X n matrices.

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A051055 'Connected' alternating sign n X n matrices, i.e., not made from smaller blocks.

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Mathematica

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A059476 Number of 4n X 4n quarter-turn symmetric alternating-sign matrices (QTSASM's).

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A107252 a(n) = Product_{k=0..n-1} (n+k)!/(k+1)!.

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Magma

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PARI

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A231498 Dimensions of algebraic generators of Hopf algebra ASM.

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A293930 Number of circularly chained n X n alternating sign matrices.

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A384062 Number of maximal antichains in the Bruhat order of type A_n.

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