A005130 -id:A005130 - cp's OEIS frontend

A143670 Array of higher spin alternating sign matrices, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 42, 26, 4, 1, 1, 429, 628, 70, 5, 1, 1, 7436, 41784, 5102, 155, 6, 1, 1, 218348, 7517457, 1507128, 28005

Offset: 1

Jonathan Vos Post, Aug 28 2008

Adapted from Table 1, p. 5: |ASM(n, r)|, where A[k,n] = |ASM(n, k)|. Abstract: We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions.

The case r = 1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are the integer points of the r-th dilate of an integral convex polytope of dimension (n-1)^2 whose vertices are the standard alternating sign matrices of size n. It then follows that, for fixed n, these matrices are enumerated by an Ehrhart polynomial in r.

			The array begins:
========================================================
....|.r=0|..r=1.|.....r=2.|.......r=3.|..........r=4.|
n=1.|..1.|...1..|......1..|.........1.|...........1..|.A000012
n=2.|..1.|...2..|......3..|.........4.|...........5..|.A000027
n=3.|..1.|...7..|.....26..|........70.|.........155..|
n=4.|..1.|..42..|....628..|......5102.|.......28005..|
n=5.|..1.|.429..|..41784..|...1507128.|....28226084..|
n=6.|..1.|7436..|7517457..|1749710096.|152363972022..|
========================================================

Roger E. Behrend and Vincent A. Knight, Higher Spin Alternating Sign Matrices, arXiv:0708.2522 [math.CO], 2007.
Roger E. Behrend, Vincent A. Knight, Higher Spin Alternating sign matrices, El. J. Combinat 14 (2007) #R83

Cf. A000012, A000027, A005130.

Apart from the trivial formulas |ASM(0, n)| = 1 (since ASM(0, n) contains only the n X n zero matrix), |ASM(1, r)| = 1 and |ASM(2, r)| = r+1, the only previously- known formula for a special case of |ASM(n, r)| is |ASM(n, 1)| = Sum_{i=0..n-1} (3*i+1)!/(n+1)!.

Some terms of the 7th diagonal from R. J. Mathar, Mar 04 2010

A180349 Gog words avoiding the subpattern 312.

Original entry on oeis.org

1, 2, 6, 26, 162, 1450, 18626, 343210, 9069306, 343611106, 18662952122, 1453016097506, 162144482866166, 25932885879826066

Offset: 1

Arvind Ayyer, Jan 18 2011

Gog words of size n are words of length n in an alphabet of odd-sized tuples of increasing integers that satisfy the following conditions:
(1) The length of the word is n,
(2) each letter in the word has maximum entry at most n,
(3) an integer in an even-numbered position in a tuple is repeated in another tuple to its left and to its right in odd-numbered positions,
(4) every repeated integer alternates in odd- and even-numbered positions in subsequent tuples.
They are in natural bijection with alternating sign matrices.
Further, the integers c, a, b form a 312-subpattern of the Gog word w = x_1 x_2 ... x_n if the following conditions hold:
(1) c, a, b appear in odd positions in x_i, x_j, x_k, respectively, where i < j < k,
(2) b is not in an even position in x_(i+1), ..., x_(k-1),
(3) if x_j = (p_1, q_1, ..., p_(k-1), q_(k-1), p_k), either b > p_k or p_l < b < q_l for some l.
(4) a < b < c.
a(n) is equal to the number of gapless Gog triangles of size n, and also to the number of gapless Magog triangles of size n. - Ludovic Schwob, May 18 2024

			For n=3, there are 7 Gog words: (1)(2)(3), (1)(3)(2), (2)(1)(3), (2)(3)(1), (3)(1)(2), (3)(2)(1) and (2)(123)(2). Of these, all but (3)(1)(2) avoid the subpattern 312.
More complicated examples: 31(234)3 and 25(12356)542 contain the subpattern 312 but 25(12456)532 does not.

Arvind Ayyer, Robert Cori, and Dominique Gouyou-Beauchamps, Monotone triangles and 312 pattern avoidance, arXiv:1101.1666 [math.CO], 2011.
Mathilde Bouvel, Rebecca Smith, and Jessica Striker, Key-avoidance for alternating sign matrices, arXiv:2408.05311 [math.CO], 2024. See p. 4.
Ludovic Schwob, Sage program.

Cf. A005130, A000108, A116715, A116722, A116735.

a(13)-a(14) from Ludovic Schwob, May 18 2024

A229165 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 and there are no adjacent -1's or 1's in any row or column.

Original entry on oeis.org

1, 2, 7, 104, 7255, 2280064, 2708080414, 13039886441460

Offset: 1

Eric W. Weisstein, Sep 25 2013

Cf. A005130 (Robbins numbers).

Cf. A050204 (entries in each row or column sum to 1).

Additional terms from R. H. Hardin, Sep 25 2013

A231499 Dimensions of totally primitive elements of Hopf algebra ASM.

Original entry on oeis.org

0, 1, 0, 2, 20, 277, 5776, 188900, 9980698, 868571406, 125895356788, 30578409431921, 12487049992168856, 8588968244025278785, 9960996172657038782698, 19489102605171036723471036, 64349079323905130216144033784, 358611512621756217468022899213127

Offset: 0

N. J. A. Sloane, Nov 09 2013

nonn

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/ASM(x) - 1/ASM(x)^2 where ASM(x) is the g.f. of A005130. - Ludovic Schwob, Feb 14 2024

More terms from Ludovic Schwob, Feb 14 2024

A293931 Number of circularly chained n-tuples of 2 X 2 alternating sign matrices.

Original entry on oeis.org

2, 10, 14, 42, 82, 214, 478, 1186, 2786

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, A293930, A059475.

A321511 Number of alternating sign n X n matrices excluding permutation matrices.

Original entry on oeis.org

0, 0, 0, 1, 18, 309, 6716, 213308, 10809896, 911472580, 129530643900, 31095704935575, 12611311380675900, 8639383512070631700, 9995541355360989190800, 19529076234659969430529200, 64427185703425668434106855840, 358869201916137601091798728321296

Offset: 0

Wendy Appleby, Nov 11 2018

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..93
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 8.

Cf. A005130.

a[n_] := Product[(3 k + 1)!/(n + k)!, {k, 0, n-1}] - n!; a /@ Range[0, 17] (* Giovanni Resta, Nov 20 2018 *)

a(n) = A005130(n) - n!

A102610 Triangle read by rows: coefficients of characteristic polynomials of lower triangular matrix of Robbins triangle numbers.

Original entry on oeis.org

0, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -11, 33, -37, 14, 1, -53, 495, -1423, 1568, -588, 1, -482, 23232, -213778, 612035, -673260, 252252, 1, -7918, 3607384, -172966930, 1590265243, -4551765520, 5006613612, -1875745872, 1, -226266, 1732486848, -787838048562, 37768573496883, -347235787044084

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Jan 30 2005

Roots of n-th characteristic polynomial are the first n Robbins numbers (A005130).

Second column of triangle is partial sums of Robbins numbers negated (A173312).

			Generation of the triangle:
We begin with A048601
1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
...
and get polynomials
x - 1
x^2 - 2*x + 1
x^3 - 4*x^2 + 5*x - 2
x^4 - 11*x^3 + 33*x^2 - 37*x + 14
x^5 - 53*x^4 + 495*x^3 - 1423*x^2 + 1568*x - 588
...

Cf. A005130, A048601.

T(n, k) = binomial(n+k-2,k-1)*((2*n-k-1)!/(n-k)!)*prod(j=0,n-2,((3*j+1)!/(n+j)!))
RM(n)=M=matrix(n,n);for(l=1,n, for(k=1,l,M[l,k]=T(l,k)));M
for(i=1,10,print(charpoly(RM(i))))

Sequence has been prepended with a(0)=0 to enable table display (so offset has been set to 0 accordingly) by Michel Marcus, Aug 23 2013

A134356 a(n) = Product_{k=1..n-1} (3k+1)!/(n+k)!.

Original entry on oeis.org

1, 4, 42, 1008, 51480, 5353920, 1100473920, 437480709120, 330886851724800, 470053968773760000, 1241242628123282400000, 6040838558884497984000000, 53797620867616662708672000000, 871394214986903051252166758400000

Offset: 1

Artur Jasinski, Oct 22 2007

nonn

Cf. A005130.

a = {}; Do[k = Product[(3i + 1)!/(n + i)!, {i, 1, n - 1}]; AppendTo[a, k], {n, 1, 20}]; a
Table[Product[(3k+1)!/(n+k)!,{k,n-1}],{n,20}] (* Harvey P. Dale, Sep 30 2015 *)

a(n) ~ Pi^(5/6) * 3^(3*n^2/2 - 7/36) * n^(n + 13/36) / (A^(1/3) * Gamma(1/3)^(2/3) * 2^(2*n^2 - 11/12) * exp(n - 1/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2017

A144017 Number of n X n X n alternating sign hypermatrices.

Original entry on oeis.org

1, 1, 2, 14, 924, 852960

Offset: 0

Samuel Zbarsky (sa_zbarsky(AT)yahoo.com), Sep 07 2008

An alternating sign hypermatrix (ASHM) of order n is an n X n X n hypermatrix with entries from {0, 1, -1} such that the nonzero entries of each row, column, and vertical line alternate in sign, beginning and ending with +1.

ASHMs are the three-dimensional analog of alternating sign matrices and generalize Latin squares, as the set of n X n X n ASHMs containing no negative entries is in bijection with the set of n X n Latin squares.

			For n = 1, the only n X n X n ASHM is [[[1]]].
For n = 2, the two n X n X n ASHMs are
[[[1,0],
  [0,1]],
 [[0,1],
  [1,0]]]
and
[[[0,1],
  [1,0]],
 [[1,0],
  [0,1]]].

R. Brualdi and G. Dahl, Alternating sign matrices and hypermatrices, and a generalization of Latin squares, Advances in Applied Mathematics, 95 (2018), 116 - 151.
Cian O'Brien, Alternating sign hypermatrix decompositions of Latin-like squares, Advances in Applied Mathematics, 121 (2020), 102097.

3-dimensional analog of A005130, generalization of A002860.

# Program written in Sage
# Returns True if a given list of n n X n ASMs form an ASHM, returns False otherwise
def ASHM(L):
    n = len(L)
    # Searches through the vertical line in position (i,j) of the hypermatrix for each i and j
    for i in range(n):
        for j in range(n):
            # Since the first nonzero entry in each line of an ASHM is +1, the alternating condition is checked
            # as if the previous nonzero entry was -1
            last = -1
            for k in range(n):
                # In each position of the current vertical line, if the sign of the current entry is the opposite
                # of the previous, then the previous sign is updated
                if L[k][i,j]*last == -1:
                    last *= -1
                # Otherwise False is returned unless the current entry is 0
                elif L[k][i,j] != 0:
                    return False
            # If the most recent nonzero entry is not +1 by the time all entries have been checked, False is returned
            if last != 1:
                return False
    # If False has not been returned, return True
    return True
# Generates all combinations of one element from each list in L
def combos(L, current = [[]]):
    # If there are no elements left which have not been combined, then return the combinations already made
    if len(L) == 0:
        return current
    # Otherwise, each element of the next list in L is appended to the current list of combinations made
    output = []
    for K in current:
        for a in L[0]:
            output.append(K + [a])
    return combos(L[1:], output)
# Counts all ASHMs of order n
def count_ASHMs(n):
    # All ASMs of order n are imported as matrices
    asms = []
    for A in AlternatingSignMatrices(n):
        asms.append(A.to_matrix())
    # Initially, zero ASHMs have been counted
    count = 0
    # Every possible combination of n n X n ASMs is checked
    for i in combos([[k for k in range(len(asms))] for m in range(n)]):
        # If the current list of n n X n ASMs forms an ASHM, then it is counted
        count += int(ASHM([asms[i[k]] for k in range(n)]))
    # The final count is returned
    return count
# Note: I ran a more efficient version of this program in Python to obtain the answer for n=5, and even then it took 6 hours.
print(count_ASHMs(0))
print(count_ASHMs(1))
print(count_ASHMs(2))
print(count_ASHMs(3))
print(count_ASHMs(4))
print(count_ASHMs(5))
# Cian O'Brien, May 31 2023

Verified using 2 computer searches. The search given counts all sequences of n alternating sign matrices of order n that form an ASHM. The other search uses corner-sum matrices, which are known to be in bijection with alternating sign matrices, by counting all 3-dimensional analogs of corner-sum matrices.

a(4) corrected and a(5) added, and definition updated by Cian O'Brien, May 31 2023

A155901 Arise in p-adic valuations of sequences counting alternating sign matrices.

Original entry on oeis.org

2, 8, 5, 12, 5, 14, 8, 14

Offset: 1

Jonathan Vos Post, Jan 30 2009

These are the values from Table 1 p.14 of Sun and Moll.

			a(7) = 8 because "the eight solutions to Nu(T(n)) = 7 are 26, 38, 46, 82, 5462, 10922, 10924 and J_15 - 1 = 21844" where J_k = k-th Jacobsthal number = A001045(k).

D. Bressoud, Proofs and Confirmations: the story of the Alternating Sign Matrix Conjecture, Cambridge University Press, 1999.

D. Bressoud and J. Propp, How the Alternating Sign Matrix Conjecture was solved, Notices Amer. Math. Soc., 46:637-646, 1999.
Xinyu Sun and Victor H. Moll, The p-adic valuations of sequences counting alternating sign matrices, arXiv:0901.4564 [math.NT], 2009.

Cf. A000219, A001045, A005130, A048601, A128445, A143670.

cp's OEIS Frontend

A143670 Array of higher spin alternating sign matrices, read by antidiagonals.

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A180349 Gog words avoiding the subpattern 312.

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A229165 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 and there are no adjacent -1's or 1's in any row or column.

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A231499 Dimensions of totally primitive elements of Hopf algebra ASM.

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A293931 Number of circularly chained n-tuples of 2 X 2 alternating sign matrices.

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A321511 Number of alternating sign n X n matrices excluding permutation matrices.

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Mathematica

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A102610 Triangle read by rows: coefficients of characteristic polynomials of lower triangular matrix of Robbins triangle numbers.

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PARI

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A134356 a(n) = Product_{k=1..n-1} (3k+1)!/(n+k)!.

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Mathematica

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A144017 Number of n X n X n alternating sign hypermatrices.

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Sage

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A155901 Arise in p-adic valuations of sequences counting alternating sign matrices.