A005130 -id:A005130 - cp's OEIS frontend

A005163 Number of alternating sign n X n matrices that are symmetric about a diagonal.

1, 2, 5, 16, 67, 368, 2630, 24376, 293770, 4610624, 94080653, 2492747656, 85827875506, 3842929319936, 223624506056156, 16901839470598576, 1659776507866213636, 211853506422044996288, 35137231473111223912310, 7569998079873075147860464

Offset: 1

N. J. A. Sloane and Simon Plouffe

Robbins's paper does not give a formula for this sequence. On the contrary he states: "Apparently these numbers do not factor into small primes, so a simple product formula seems unlikely. Of course this does not rule out other very simple formulas, but these would be more difficult to discover (let alone prove)." As far as I know no formula is currently known. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

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.

Christoph Koutschan, Table of n, a(n) for n = 1..131
Roger E. Behrend, Ilse Fischer, and Christoph Koutschan, Diagonally symmetric alternating sign matrices, arXiv:2309.08446 [math.CO], 2023.
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). Discrete Math. 139 (1995), 57-72.
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]

More terms (taken from Bousquet-Mélou & Habsieger's paper) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

A050204 a(n) is the number of n X n matrices of 0's, 1's and -1's in which the entries in each row or column sum to 1.

Original entry on oeis.org

1, 2, 21, 1344, 628080, 2030271480, 49846153939785, 9350902053107858880, 13807842729396124460629536, 162526525004284727135887319788800, 15464071313054035722739424623204673044920, 12011244510939290610945342901003078567310643860160

Offset: 1

Eric W. Weisstein

nonn

Cf. A005130 (Robbins numbers).
Cf. A229165 (entries in each row or column sum to 1 and there are no adjacent -1's or 1's in any row or column).
Cf. A172645 (case row and column sums are zero).

Definition corrected by Murphy Waggoner and R. H. Hardin, Sep 24 2013
a(6)-a(8) from R. H. Hardin, Sep 25 2013
a(9)-a(12) from Andrew Howroyd, Feb 03 2021

A293179 Number of linearly chained n-tuples of 2 X 2 alternating sign matrices.

Original entry on oeis.org

2, 17, 4, 159, 8, 1129, 16, 7151

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, A293180.

A293180 Number of linearly chained n-tuples of 3 X 3 alternating sign matrices.

Original entry on oeis.org

7, 504, 49, 98028

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.

A378072 Number of elements in the completion of the Bruhat order on B_n.

Original entry on oeis.org

1, 2, 10, 132, 4824, 493600, 142343254

Offset: 0

Ludovic Schwob, Nov 16 2024

Lascoux and Schützenberger have shown that the completion of type B Bruhat order is distributive, and have described its irreducible elements. It follows that a(n) is the number of antichains in the poset of irreducible elements of the Bruhat order on B_n.

This sequence is a type B analog of A005130, since ASMs form a lattice isomorphic to the completion of type A Bruhat order.

A. Lascoux and M.-P. Schützenberger, Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), #R27.

Cf. A005130 (ASMs), A005158 (centrally symmetric ASMs), A000165 (number of elements in B_n).

a(6) by Dmitry I. Ignatov, May 16 2025

A005161 Number of alternating sign 2n+1 X 2n+1 matrices symmetric with respect to both horizontal and vertical axes (VHSASM's).

Original entry on oeis.org

1, 1, 1, 2, 6, 33, 286, 4420, 109820, 4799134, 340879665, 42235307100, 8564558139000, 3012862604463000, 1742901718473961200, 1742218029490675762080, 2873822682985675809192288, 8167157387273280570395662320, 38402596062535617548517706584760, 310388509293255836481583597538626504

Offset: 0

N. J. A. Sloane

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.

Andrew Howroyd, Table of n, a(n) for n = 0..80
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 24.
Ira Gessel and Guoce Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
Soichi Okada, Enumeration of symmetry classes of alternating sign matrices and characters of classical groups, Journal of Algebraic Combinatorics volume 23, pages 43-69 (2006).
Pavel Pyatov, Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon, arXiv:math-ph/0406025, 2004. [_Vladeta Jovovic_, Aug 15 2008]
David P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
Richard P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire énumérative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Cf. A005130, A005162, A005163, A005156, A051255, A059476.

\\ here b(n) and c(n) are A005156 and A051255.
b(n) = prod(k=0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
c(n) = prod(k=0, n-1, (3*k+1)*(6*k)!*(2*k)!/((4*k)!*(4*k+1)!));
a(n) = b(n\2) * c((n+1)\2) \\ Andrew Howroyd, May 09 2023

Robbins gives a simple (conjectured) formula, which was proven by Okada.
a(2*n) = A005156(n)*A051255(n); a(2*n+1) = A005156(n)*A051255(n+1). - Paul Zinn-Justin, May 05 2023
a(n) = A005156(floor(n/2)) * A051255(ceiling(n/2)). - Andrew Howroyd, May 09 2023

More terms (from the P. Pyatov paper) from Vladeta Jovovic, Aug 15 2008

Terms a(13) and beyond from Andrew Howroyd, May 09 2023

A005164 Number of alternating sign 2n+1 X 2n+1 matrices invariant under all symmetries of the square.

Original entry on oeis.org

1, 1, 1, 2, 4, 13, 46, 248, 1516, 13654, 142873, 2156888, 38456356, 974936056, 29540545024, 1259111024288, 64726478396896, 4641989615977216, 404396533544588344, 48825344233129714772, 7202552030561982627472, 1464587581921220811285325, 365627222082497915618219716, 125253905685915522767942493032, 52893528399758443649956432899616

Offset: 0

N. J. A. Sloane and Simon Plouffe

M. Bousquet-Mélou and L. Habsieger, Sur les matrices à signes alternants, Séries Formelles et Combinatoire Algébrique, 4th colloquium, 15-19 Juin 1992, Montréal, Université du Québec à Montréal, pp. 19-32.
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.

C. Hagendorf and J. Liénardy The open XXZ chain at ∆ = -1/2 and the boundary quantum Knizhnik-Zamolodchikov equations, arXiv:2008.03220 [math-ph], 2020.
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]

Cf. A005130.

Hagendorf and Liénardy give a (conjectured) formula in terms of multiple contour integrals. - Jean Liénardy, Aug 15 2020

a(14)-a(19) from Jean Liénardy, Aug 15 2020

a(20)-a(24) from Jean Liénardy, Sep 21 2022

A059477 3-enumeration of n X n alternating-sign matrices.

Original entry on oeis.org

1, 1, 2, 9, 90, 2025, 102060, 11573604, 2946308904, 1687603650084, 2171945897658108, 6289412333143466241, 40940643700218614247324, 599627833263501883888374756, 19747212169938041691404746667280, 1463229065460461810019231236067824400

Offset: 0

N. J. A. Sloane, Feb 04 2001

nonn

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices, arXiv:math-ph/0404045, 2004.
F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices, Advances in Applied Mathematics 34 (2005) 798.
F. Colomo and A. G. Pronko, Square ice, alternating sign matrices and classical orthogonal polynomials, arXiv:math-ph/0411076, 2004; JSTAT (2005) P01005.
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.
Yu. G. Stroganov, 3-enumerated alternating sign matrices, arXiv:math-ph/0304004, 2003.

Cf. A005130.

A059477 := proc(n) local i,j,t1; t1 := 3^(n^2-n)*2^(-n^2+n); for i from 1 to n do for j from 1 to n do if j-i mod 2 <> 0 then t1 := t1*(3*j-3*i+1)/(3*j-3*i); fi; od; od; t1; end;

a[0] = 1; a[n_?OddQ] := a[n] = 3^((1/2)*((n-1)/2 + 1)*(n-1)) * Product[(3*k - 1)!^2/(k + (n-1)/2)!^2, {k, 1, (n - 1)/2}];
a[n_?EvenQ] := (3^((n-2)/2)*((3*(n-2))/2 + 2)!*((n - 2)/2)! * a[n - 1])/(n - 1)!^2;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 28 2017, after Ralf Stephan *)

a(2m+1)=3^(m*(m+1))*prod(k=1, m, ((3*k-1)!/(m+k)!)^2), a(2m+2)=3^m*(3*m+2)!*m!/((2*m+1)!)^2*a(2m+1). - Ralf Stephan, Apr 24 2004

A128445 Number of facets of the Alternating Sign Matrix polytope ASM(n).

Original entry on oeis.org

1, 1, 2, 8, 20, 40, 68, 104, 148, 200, 260, 328, 404, 488, 580, 680, 788, 904, 1028, 1160, 1300, 1448, 1604, 1768, 1940, 2120, 2308, 2504, 2708, 2920, 3140, 3368, 3604, 3848, 4100, 4360, 4628, 4904, 5188, 5480, 5780, 6088, 6404, 6728, 7060, 7400, 7748, 8104

Offset: 0

Jonathan Vos Post, May 09 2007

The number of vertices (Bressoud) is Product_{j=0..n-1}(3j+1)!/(n+j)!.

D. M. Bressoud, Proofs and confirmations: the story of the alternating sign matrix conjecture, MAA Spectrum, 1999.

Jessica Striker, The alternating sign matrix polytope, arXiv:0705.0998 [math.CO], 2007-2009.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005130 (number of vertices).

Table[4((n-2)^2+1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{20,8,4},50] (* Harvey P. Dale, Mar 05 2012 *)

a(n)=([0,1,0; 0,0,1; 1,-3,3]^n*[20;8;4])[1,1] \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*((n-2)^2 + 1) for n >= 3.
From Harvey P. Dale, Mar 05 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n > 5.
G.f.: (2*x^5+x^4+4*x^3+2*x^2-4*x+1)/(1-x)^3. (End)

More terms from Harvey P. Dale, Mar 05 2012

Initial 3 terms and formulas corrected by Ludovic Schwob, Feb 14 2024

A384061 Number of antichains in the Bruhat order of type A_n.

Original entry on oeis.org

3, 9, 250, 67595432

Offset: 1

Dmitry I. Ignatov, May 18 2025

The number of 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 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 antichains is {{}, {1}, {s2}, {s2, s1}, {s1}, {s2*s1}, {s2*s1, s1*s2}, {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), A384062.

cp's OEIS Frontend

A005163 Number of alternating sign n X n matrices that are symmetric about a diagonal.

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A050204 a(n) is the number of n X n matrices of 0's, 1's and -1's in which the entries in each row or column sum to 1.

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A293179 Number of linearly chained n-tuples of 2 X 2 alternating sign matrices.

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A293180 Number of linearly chained n-tuples of 3 X 3 alternating sign matrices.

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A378072 Number of elements in the completion of the Bruhat order on B_n.

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A005161 Number of alternating sign 2n+1 X 2n+1 matrices symmetric with respect to both horizontal and vertical axes (VHSASM's).

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PARI

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A005164 Number of alternating sign 2n+1 X 2n+1 matrices invariant under all symmetries of the square.

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A059477 3-enumeration of n X n alternating-sign matrices.

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Maple

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A128445 Number of facets of the Alternating Sign Matrix polytope ASM(n).

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Programs

Mathematica

PARI

Formula

Extensions

A384061 Number of antichains in the Bruhat order of type A_n.

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