A005130 -id:A005130 - cp's OEIS frontend

A112332 a(n) = Product_{k=0..n-1} k!*binomial(2k,k).

1, 1, 2, 24, 2880, 4838400, 146313216000, 97339256340480000, 1683704371913057894400000, 873705178746128941669416960000000, 15414977576506278044562764045746176000000000, 10334857226047177887548812577909403133201612800000000000

Offset: 0

Paul Barry, Sep 04 2005

Cf. A005130, A110131.

seq(mul(mul((j+k),j=1..k), k=1..n), n=-1..9); # Zerinvary Lajos, Sep 21 2007

Table[Product[(2*k)!/k!,{k,0,n-1}],{n,0,10}] (* Vaclav Kotesovec, Jul 11 2015 *)

a(n)=denominator(Product{k=0..n-1, (2k+1)!/(n+k)!}).
G.f.: 1+ x*G(0)/2, where G(k)= 1  + 1/(1 - 1/(1 + 1/((2*k+2)!/(k+1)!)/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
a(n) ~ A^(1/2) * 2^(n^2 - n/2 - 7/24) * n^(n^2/2 - n/2 + 1/24) / exp(3*n^2/4 - n/2 + 1/24), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 11 2015
From Alois P. Heinz, Jun 30 2022: (Start)
a(n) = Product_{i=1..n-1} Product_{j=i..n-1} (i+j).
a(n) = A110131(n). (End)

A048601 Robbins triangle read by rows: T(n,k) = number of alternating sign n X n matrices with a 1 at top of column k (n >= 1, 1<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 7, 14, 14, 7, 42, 105, 135, 105, 42, 429, 1287, 2002, 2002, 1287, 429, 7436, 26026, 47320, 56784, 47320, 26026, 7436, 218348, 873392, 1813968, 2519400, 2519400, 1813968, 873392, 218348, 10850216, 48825972, 113927268, 179028564

Offset: 1

N. J. A. Sloane

An alternating sign matrix is a matrix of 0's and 1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.

Named after the American mathematician David Peter Robbins (1942-2003). - Amiram Eldar, Jun 13 2021

			Triangle begins:
     1,
     1,     1,
     2,     3,     2,
     7,    14,    14,     7,
    42,   105,   135,   105,    42,
   429,  1287,  2002,  2002,  1287,   429,
  7436, 26026, 47320, 56784, 47320, 26026, 7436,
  ...

David Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, 1999, p. 5.

N. J. A. Sloane, Table of n, a(n) for n = 1..1275 [Rows 1..50, flattened]
Roger E. Behrend, Philippe Di Francesco and Paul Zinn-Justin, On the weighted enumeration of Alternating Sign Matrices and Descending Plane Partitions, arXiv:1103.1176 [math.CO], 2011.
David Bressoud and James Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., Vol. 46, No. 6 (1999), p. 637-646.
FindStat - Combinatorial Statistic Finder, The column of the unique '1' in the first row of the alternating sign matrix.
FindStat - Combinatorial Statistic Finder, The column of the unique 1 in the first row of the alternating sign matrix.
P. Di Francesco, A refined Razumov-Stroganov conjecture II, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.
D. Gerdemann, Robbins Triangle for Counting Alternating Sign Matrices YouTube Video, 2015.
W. H. Mills, David P. Robbins and Howard Rumsey, Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory, Ser. A, Vol. 34, No. 3 (1983), pp. 340-359. MR0700040 (85b:05013).
Eric Weisstein's World of Mathematics, Alternating Sign Matrix.
Doron Zeilberger, Proof of the Refined Alternating Sign Matrix Conjecture, arXiv:math/9606224 [math.CO], 1996.
Doron Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math., Vol. 34 (2005), pp. 939-954.

Row sums (also borders) of triangle give A005130. Cf. A051106.

A210697 is a companion triangle.

T:=(n,k)-> binomial(n+k-2, k-1)*((2*n-k-1)!/(n-k)!)*product(((3*j+1)!/(n+j)!),j=0..n-2);

t[n_, k_] := Binomial[n+k-2, k-1]*((2*n-k-1)!/(n-k)!)*Product[((3*j+1)!/(n+j)!), {j, 0, n-2}]; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 12 2012, from formula *)

T(n,k) = binomial(n+k-2, k-1)*((2*n-k-1)!/(n-k)!) * product(((3*j+1)!/(n+j)!), j=0..n-2);

More terms from James Sellers

A005157 Number of totally symmetric plane partitions that fit in an n X n X n box.

Original entry on oeis.org

1, 2, 5, 16, 66, 352, 2431, 21760, 252586, 3803648, 74327145, 1885102080, 62062015500, 2652584509440, 147198472495020, 10606175914819584, 992340657705109416, 120567366227960791040, 19023173201224270401428, 3897937005297330777227264

Offset: 0

N. J. A. Sloane

Also, number of 2-dimensional shifted complexes on n+1 nodes. [Klivans]
Also the number of totally symmetric partitions which fit in an (n-1)-dimensional box with side length 4 (for n>0). - Graham H. Hawkes, Jan 11 2014
Suppose we index this sequence slightly differently. Let the elements of a partition be represented by points rather than boxes, as in a Ferrers diagram. In this case, a 1 X 1 X 1 (closed) box would fit 8 points -- one at each vertex of the box, and we use the convention that a 0 X 0 X 0 (closed) box contains exactly one point. Using this indexing, the sequence begins (offset is still 0) 2,5,16,... rather than 1,2,5,... If we use the same method of indexing for all other dimensions, then we have the following remarkable result: The number of totally symmetric partitions which fit inside a d-dimensional box with side length n is equal to the number of totally symmetric partitions which fit inside an n-dimensional box of side length d. - Graham H. Hawkes, Jan 11 2014
For two other contexts where this sequence arises, see the Knuth (2019) link (noncrossing paths among the 2(2^n-1) paths defined in that note; independent sets of paths among the first 2^n-1 of those). - N. J. A. Sloane, Feb 09 2019, based on email from Don Knuth.

			a(2) = 5 because we have: void, 1, 21/1, 22/21, and 22/22.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198 (corrected).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..130 (first 51 terms from T. D. Noe)
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
R. K. Guy, Letter to N. J. A. Sloane, Dec 5 1988.
Graham H. Hawkes, Totally symmetric partitions in boxes
Seth Ireland, A bijection between strongly stable and totally symmetric partitions, arXiv:2302.02505 [math.CO], 2023.
C. Klivans, Obstructions to shiftedness, preprint.
C. Klivans, Obstructions to shiftedness, Discrete Comput. Geom., 33 (2005), 535-545.
Don Knuth, A conjecture about noncrossing paths, Feb 06 2019.
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.
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. A049505, A184173, A323848.

Row sums of A214564.

A005157 := proc(n) local i,j; mul(mul((i+j+n-1)/(i+2*j-2),j=i..n),i=1..n); end;

Table[Product[(i+j+k-1)/(i+j+k-2),{i,n},{j,i,n},{k,j,n}],{n,0,20}] (* Harvey P. Dale, Jul 17 2011 *)

A005157(n)=prod(i=1,n,prod(j=i,n,(i+j+n-1)/(i+2*j-2))) \\ M. F. Hasler, Sep 26 2018

a(n) = Product_{i=1..n} Product_{j=i..n} Product_{k=j..n} (i+j+k-1)/(i+j+k-2). - Paul Barry, May 13 2008
a(n) ~ exp(1/72) * GAMMA(1/3)^(2/3) * n^(7/72) * 3^(3*n*(n+1)/4 + 11/72) / (A^(1/6) * Pi^(1/3) * 2^(n*(2*n+1)/2 + 13/24)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015
a(n) = sqrt(A323848(n+1,n)) for n >= 1. [proof by Nikolai Beluhov; see Knuth (2019) link] - Alois P. Heinz, Feb 10 2019
Apparently, a(n) = Sum_{k=0..n} A184173(n,k). - Alois P. Heinz, Feb 11 2019
Conjectures: if p == 1 (mod 6) is prime then a(p) == 2^((p+5)/6) (mod p^2); if p == 5 (mod 6) is prime then a(p) == 2^((p+1)/6) (mod p^2) (checked up to p = 1009). - Peter Bala, Feb 17 2023

A005158 Number of alternating sign n X n matrices invariant under a half-turn.

Original entry on oeis.org

1, 2, 3, 10, 25, 140, 588, 5544, 39204, 622908, 7422987, 198846076

Offset: 1

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.

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

A059475(n) = a(2n).

Robbins gives simple (conjectured) formulas related to this sequence in Section 3.3.

a(n) = a(n-1) * (1 + [n even]/3) * C(n\2*3, n\2) / C(n\2*2, n\2) for all n > 1, where C(.,.) are the binomial coefficients, n\2 := floor(n/2) and [n even] = 1 if n is even, 0 else (Iverson bracket). [From Robbins conjectured(!) formulas.] - M. F. Hasler, Jun 15 2019

A006366 Number of cyclically symmetric plane partitions in the n-cube; also number of 2n X 2n half-turn symmetric alternating sign matrices divided by number of n X n alternating sign matrices.

Original entry on oeis.org

1, 2, 5, 20, 132, 1452, 26741, 826540, 42939620, 3752922788, 552176360205, 136830327773400, 57125602787130000, 40191587143536420000, 47663133295107416936400, 95288872904963020131203520, 321195665986577042490185260608

Offset: 0

N. J. A. Sloane

In the 1995 Encyclopedia of Integer Sequences this sequence appears twice, as both M1529 and M1530.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.7) on page 198, except the formula given is incorrect. It should be as shown here.
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.

Vincenzo Librandi, Table of n, a(n) for n = 0..90
G. E. Andrews,Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225.
P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.
W. F. Lunnon, The Pascal matrix, Fib. Quart. vol. 15 (1977) pp. 201-204.
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]
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.

Cf. A005130, also A003827, A005156, A005158, A005160-A005164, A048601, A050204.

A006366 := proc(n) local i, j; mul((3*i - 1)*mul((n + i + j - 1)/(2*i + j - 1), j = i .. n)/(3*i - 2), i = 1 .. n) end;

Table[Product[(3i-1)/(3i-2) Product[(n+i+j-1)/(2i+j-1),{j,i,n}],{i,n}],{n,0,20}] (* Harvey P. Dale, Apr 17 2013 *)

a(n)=prod(i=0,n-1,(3*i+2)*(3*i)!/(n+i)!)

a(n) = Product_{i=1..n} (((3*i-1)/(3*i-2))*Product_{j=i..n} (n+i+j-1)/(2*i+j-1)).

a(n) ~ exp(1/36) * GAMMA(1/3)^(4/3) * n^(7/36) * 3^(3*n^2/2 + 11/36) / (A^(1/3) * Pi^(2/3) * 2^(2*n^2 + 7/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A059475 Number of 2n X 2n half-turn symmetric alternating-sign matrices (HTSASM's).

Original entry on oeis.org

1, 2, 10, 140, 5544, 622908, 198846076, 180473355920, 465904151957920, 3422048076740462480, 71525763221287897903500, 4254840960508487045451825000, 720428791920558617462950575000000, 347230535542092373572967034254050000000

Offset: 0

N. J. A. Sloane, Feb 04 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..66
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
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.
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2001.

Even-numbered terms of A005158.

Cf. A005130, A006366, A049503.

a[n_] := Product[(3k+1)(3k+2)(3k)!^2/(n+k)!^2, {k, 0, n-1}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Sep 01 2018, after Seiichi Manyama *)

a(n) = A005130(n)*A006366(n).
a(n) = A049503(n)*Product_{k=0..n-1} (3*k+2)/(3*k+1). - Seiichi Manyama, Jul 29 2018
a(n) ~ exp(1/18) * Gamma(1/3)^(2/3) * n^(1/18) * 3^(3*n^2 + 1/9) / (A^(2/3) * Pi^(1/3) * 2^(4*n^2 + 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 26 2020

A297622 Triangle read by rows: a(n,k) is the number of k X n matrices which are the first k rows of an alternating sign matrix of size n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 7, 7, 1, 4, 16, 42, 42, 1, 5, 30, 149, 429, 429, 1, 6, 50, 406, 2394, 7436, 7436, 1, 7, 77, 938, 9698, 65910, 218348, 218348, 1, 8, 112, 1932, 31920, 403572, 3096496, 10850216, 10850216, 1, 9, 156, 3654, 90576, 1931325, 29020904, 247587252, 911835460, 911835460

Offset: 0

Ben Branman, Jan 01 2018

Comments: An alternating sign matrix of size n is an n X n matrix of 0's, 1's and -1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.  If k < n, we relax the condition on the columns slightly, and require that
(a) If a column is not all zeros, the first nonzero entry is 1;
(b) The nonzero entries in each column alternate in sign.
The second reference gives a sequence of partially ordered sets Phi_n such that the alternating sign matrices of size n are in bijection with the maximal chains of Phi_n.  This sequence gives the number of saturated chains in Phi_n which begin at the root vertex and end at any vertex of height k.

			a(3,3)=7 because there are seven alternating sign matrices of size 3.  Six of these are the permutation matrices, and the seventh is the matrix ((0,1,0),(1,-1,1),(0,1,0)).
a(n,0)=1 because there is only one possible n X 0 matrix: the empty matrix.
a(4,4)=42 because there are 42 4 X 4 alternating sign matrices.  If we look only at the first two rows in each of the 42 4 X 4 alternating sign matrices, we get 16 distinct 2 X 4 matrices, and so a(4,2)=16.  The 16 2 X 4 matrices are
  {{0,  0,  0,  1}, {0,  0,  1,  0}},
  {{0,  0,  0,  1}, {0,  1,  0,  0}},
  {{0,  0,  0,  1}, {1,  0,  0,  0}},
  {{0,  0,  1,  0}, {0,  0,  0,  1}},
  {{0,  0,  1,  0}, {0,  1,  0,  0}},
  {{0,  0,  1,  0}, {1,  0,  0,  0}},
  {{0,  1,  0,  0}, {0,  0,  0,  1}},
  {{0,  1,  0,  0}, {0,  0,  1,  0}},
  {{0,  1,  0,  0}, {1,  0,  0,  0}},
  {{1,  0,  0,  0}, {0,  0,  0,  1}},
  {{1,  0,  0,  0}, {0,  0,  1,  0}},
  {{1,  0,  0,  0}, {0,  1,  0,  0}},
  {{0,  0,  1,  0}, {0,  1, -1,  1}},
  {{0,  0,  1,  0}, {1,  0, -1,  1}},
  {{0,  1,  0,  0}, {1, -1,  0,  1}},
  {{0,  1,  0,  0}, {1, -1,  1,  0}}.
Triangle begins:
=============================================================================================
n\k|  0  1   2    3      4       5         6          7           8            9           10
---|-----------------------------------------------------------------------------------------
_0 |  1
_1 |  1  1
_2 |  1  2   2
_3 |  1  3   7    7
_4 |  1  4  16   42     42
_5 |  1  5  30  149    429     429
_6 |  1  6  50  406   2394    7436      7436
_7 |  1  7  77  938   9698   65910    218348     218348
_8 |  1  8 112 1932  31920  403572   3096496   10850216    10850216
_9 |  1  9 156 3654  90576 1931325  29020904  247587252   911835460    911835460
10 |  1 10 210 6468 229680 7722110 205140540 3586953760 33631201864 129534272700 129534272700
  ...

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999

Robert G. Wilson v, Table of n, a(n) for n = 0..104
Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 33.
Paul Terwilliger, A Poset Phi_n whose maximal chains are in bijection with the n by n alternating sign matrices, arXiv:1710.04733 [math.CO], 2017.

Cf. A005130.

(* First we compute the Hasse diagram for Terwilliger's poset as a directed graph object. *)
ToAlternatingSignList[list_] :=
Module[{s = 1},
  Table[If[list[[k]] == 0, 0, (s = -s); -s], {k, 1, Length[list]}]]
AllAlternatingSignRows[n_] :=
AllAlternatingSignRows[
   n] = (ToAlternatingSignList /@
    Select[Table[IntegerDigits[q, 2, n], {q, 0, 2^n - 1}],
     OddQ[Total[#]] &])
output[vertex_] :=
Select[Table[
   vertex + li, {li, AllAlternatingSignRows[Length[vertex]]}],
  And[Min[#] >= 0, Max[#] <= 1] &]
elist[vertex_] := ((vertex \[DirectedEdge] #) & /@ output[vertex])
ASMPoset[n_] :=
ASMPoset[n] =
  Graph[Flatten[
    Table[elist[IntegerDigits[k, 2, n]], {k, 0, 2^n - 1}]]]
(*Now we compute the number of paths of length k starting at the root vertex.*)
ASMPosetAdjacencyMatrix[n_] := Normal[AdjacencyMatrix[ASMPoset[n]]]
Table[Total /@
  First /@ NestList[ASMPosetAdjacencyMatrix[n].# &,
    IdentityMatrix[2^n], n], {n, 1, 10}]

a(n,0) = 1;
a(n,1) = n;
a(n,n-1) = a(n,n) = A005130(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.

A003827 'Core' alternating sign n X n matrices, i.e., those that are not 'blown up' from a smaller matrix by inserting row i, column j with a_ij = 1 and all other entries in that row and column equal to 0.

Original entry on oeis.org

1, 2, 59, 1292, 53862, 3615208, 392961340, 68986099580, 19595297946515, 9048133666290540, 6832278662513786160, 8489106538840284343800, 17456177529017536829265000, 59700294731704834466701403040, 340945552945616104095546549396336, 3261527521637774696821080128931389072

Offset: 3

Christine Bessenrodt

nonn

R. K. Guy, Unsolved Problems in Number Theory, D1.

Alois P. Heinz, Table of n, a(n) for n = 3..100

Cf. A051055, A005130.

\\ rather inefficient, should use memoization
b(n) = prod(i=0, n-1, (3*i+1)!/(n+i)! );
a(n) = b(n) - n! - sum(k=1, n-3, binomial(n,k)^2 * k! *a(n-k) );
vector(20,n,a(n)) \\ Joerg Arndt, Oct 03 2015

Let b(n) = Product_{i=0..n-1} (3*i+1)!/(n+i)! be the number of alternating sign n X n matrices (i.e., sequence A005130), and a(n) the number of core alternating sign n X n matrices considered here, with the sequence [1,2,59,...] starting at offset n=3. Then it is not hard to show that for n>3: a(n) = b(n) - n! - Sum_{k=1..n-3} binomial(n,k)^2 * k! *a(n-k). - Christine Bessenrodt, Oct 02 2015

a(n) ~ exp(1/36) * Pi^(1/3) * 2^(5/12 - 2*n^2) * 3^(-7/36 + 3*n^2/2) / (A^(1/3) * Gamma(1/3)^(2/3) * n^(5/36)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 25 2016

Corrected and extended by Christine Bessenrodt, Oct 02 2015

A293178 Number of linearly chained pairs of n X n alternating sign matrices.

Original entry on oeis.org

2, 17, 504, 53932

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.

A005160 Number of alternating sign n X n matrices invariant under a quarter turn.

Original entry on oeis.org

1, 0, 1, 2, 3, 0, 12, 40, 100, 0, 1225, 6860, 28812, 0, 1037232, 9779616

Offset: 1

N. J. A. Sloane

Robbins incorrectly gives a(12) = 6460.

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.

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

a(4n) gives A059476.

Robbins gives a simple (conjectured) formula.

cp's OEIS Frontend

A112332 a(n) = Product_{k=0..n-1} k!*binomial(2k,k).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A048601 Robbins triangle read by rows: T(n,k) = number of alternating sign n X n matrices with a 1 at top of column k (n >= 1, 1<=k<=n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A005157 Number of totally symmetric plane partitions that fit in an n X n X n box.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A005158 Number of alternating sign n X n matrices invariant under a half-turn.

Views

Author

Keywords

References

Links

Crossrefs

Formula

A006366 Number of cyclically symmetric plane partitions in the n-cube; also number of 2n X 2n half-turn symmetric alternating sign matrices divided by number of n X n alternating sign matrices.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A059475 Number of 2n X 2n half-turn symmetric alternating-sign matrices (HTSASM's).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A297622 Triangle read by rows: a(n,k) is the number of k X n matrices which are the first k rows of an alternating sign matrix of size n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula