A005130 -id:A005130 - cp's OEIS frontend

A180512 Triangle of the number of alternating sign matrices according to the number of -1's.

1, 2, 6, 1, 24, 16, 2, 120, 200, 94, 14, 1, 720, 2400, 2684, 1284, 310, 36, 2, 5040, 29400, 63308, 66158, 38390, 13037, 2660, 328, 26, 1

Offset: 1

Arvind Ayyer, Jan 20 2011

The first column is the factorial, A000142.
The second column forms coefficients of Laguerre polynomials, A001810.
From Arvind Ayyer, Mar 15 2018: (Start)
Consider the row generating function A_n(x) = sum_k a(n,k) x^k. Then
A_n(0) = n!, A000142.
A_n(1) = number of ASM's, A005130.
A_n(2) = number of domino tilings of the Aztec diamond, A006125.
A_n(3) = 3-enumeration of n X n alternating-sign matrices, A059477. (End)

			In triangular format, the numbers of ASMs is as follows:
n=1:1
n=2:2
n=3:6,1
n=4:24,16,2
n=5:120,200,94,14,1
n=6:720,2400,2684,1284,310,36,2
n=7:5040,29400,63308,66158,38390,13037,2660,328,26,1

FindStat - Combinatorial Statistic Finder, The number of entries equal to negative one in the alternating sign matrix
Florent Le Gac, Quelques problèmes d’énumération autour des matrices à signes alternants, thesis, LaBRI Bordeaux, 2011.
Wikipedia, Alternating Sign Matrix

Row sums are A005130

Cf. A000142, A006125, A059477, A001810.

T(7, 7) corrected by Arvind Ayyer, Feb 12 2018

A293932 Number of circularly chained n-tuples of 3 X 3 alternating sign matrices.

Original entry on oeis.org

20, 140, 3861, 7436

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, A293931.

A306397 Sum of the coefficients in the Schur expansion of Product_{1<=i<=j<=n} (1+x_i+x_j), which is the total Chern class for the vector bundle Sym^2(E) where E is a vector bundle over a smooth complex projective variety of rank n with Chern roots x_1,...,x_n.

Original entry on oeis.org

3, 16, 147, 2304, 61347, 2768896, 211579212, 27349221376, 5977081440300, 2207706749337600, 1377785820669766875

Offset: 1

Sara Billey, Feb 12 2019

Also, the sum of 2^{number of 1's in T} summed over all reverse flagged fillings T of partition shape contained in (n,n-1,...,1) with row i bounded by n-i.

S. Billey, B. Rhoades, and V. Tewari, Boolean product polynomials, Schur positivity, and Chern plethysm, arXiv:1902.11165 [math.CO], 2019.
A. Lascoux, Classes de Chern d'un produit tensoriel, C. R. Acad. Sci. Paris Ser. A-B286 (1978), 385--387.

Cf. A005130.

A342181 Product of first n Robbins numbers.

Original entry on oeis.org

1, 1, 2, 14, 588, 252252, 1875745872, 409565359659456, 4443872618422784042496, 4052080633200943761869999708160, 524883317743439723147432404145717855232000, 16321637725818077271987866314412476606229589461376000000

Offset: 0

Vaclav Kotesovec, Mar 04 2021

nonn

Eric Weisstein's World of Mathematics, Alternating Sign Matrix.
Wikipedia, Barnes G-function.

Cf. A005130.

b:= proc(n) option remember; `if`(n<2, 1, b(n-1)*
      (n-1)!*(3*n-2)!/((2*n-2)!*(2*n-1)!))
    end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..12);  # Alois P. Heinz, Mar 04 2021

Table[Product[Product[(3*j + 1)!/(k + j)!, {j, 0, k-1}], {k,1,n}], {n,0,12}]
FoldList[Times, 1, Table[Product[(3*j + 1)!/(n + j)!, {j, 0, n - 1}], {n, 1, 12}]]

a(n) = Product_{k=1..n} A005130(k).

a(n) ~ Pi^(n/3 + 1/6) * 3^(n^3/2 + 3*n^2/4 + n/18 - 13/216) * exp(n/6 + 11*zeta(3)/(144*Pi^2) + 19/216) / (BarnesG(1/3)^(2/3) * n^(5*n/36 + 5/72) * 2^(2*n^3/3 + n^2 - n/12 - 1/12) * A^(n/3 + 19/18) * Gamma(1/3)^(2*n/3 + 7/9)), where A is the Glaisher-Kinkelin constant A074962.

A363685 Irregular triangle read by rows: T(n,k) is the number of descending plane partitions of order n with the sum of the parts equal to k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 3, 4, 4, 7, 7, 10, 11, 14, 14, 18, 18, 21, 21, 23, 22, 25, 22, 23, 21, 21, 18, 18, 14, 14, 11, 10, 7, 7, 4, 4, 3, 2, 1, 1, 0, 1

Offset: 1

Ludovic Schwob, Jun 15 2023

			Rows 1 through 5 are
  1;
  1, 0, 1;
  1, 0, 1, 1, 1, 1, 1, 0, 1;
  1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1;
  1, 0, 1, 1, 2, 3, 4, 4, 7, 7, 10, 11, 14, 14, 18, 18, 21, 21, 23, 22, 25, 22, 23, 21, 21, 18, 18, 14, 14, 11, 10, 7, 7, 4, 4, 3, 2, 1, 1, 0, 1.

Row sums give A005130.

Cf. A198890, A005130, A064999.

T(0, 0) = 1, T(1, 0) = 1, T(n, k) = 0 for k < 0 or k > (1/3)*(n+1)*n*(n-1).

A363689 Number of alternating sign matrices of size n which are indecomposable by direct and skew sums.

Original entry on oeis.org

1, 1, 0, 1, 16, 257, 5636, 187432, 9956054, 867875816, 125861715682, 30575624363575, 12486657463326416, 8588874417105963246, 9960958211434474577452, 19489076635898035234022220, 64349049302453152886635937502, 358611453994361731825817399132622, 3373425033361375072576454177693810076

Offset: 0

Ludovic Schwob, Jun 15 2023

nonn

			a(3) = 1 because the only ASM of size 3 indecomposable by direct and skew sums is
  [0  1 0]
  [1 -1 1]
  [0  1 0]

Cf. A005130, A231498, A359856.

G.f.: 2*(2 - 1/ASM(x)) - ASM(x) where ASM(x) is the g.f. of A005130.

A366449 Number of smooth discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n} having neutral element/absorbing element.

Original entry on oeis.org

2, 5, 18, 102, 970, 15947, 453872, 22174642, 1846384884, 260939482721, 62454382216334, 25285347265901814, 17304115945924822724, 20008412370393070905186, 39078178288867371807316956, 128893469663525965017925474046, 717867336460661639426421067202992, 6750439274904330523572066561554305664

Offset: 1

Marc Munar, Oct 10 2023

The number of smooth discrete aggregation functions on the finite chain L_n={0,1,...,n-1,n} having neutral element/absorbing element e\in L_n, i.e., the number of monotonic increasing binary functions F: L_n^2->L_n such that F(0,0)=0 and F(n,n)=n (discrete aggregation function); F(x+1,y)-F(x,y)<=1 and F(x,y+1)-F(x,y)<= 1 (smooth); and F(x,a)=F(a,x)=x (neutral element) or F(x,a)=F(a,x)=a (absorbing element).

Cf. A005130.

Table[2*Product[Factorial[3 i + 1]/Factorial[n + i], {i, 0, n - 1}] +
  Sum[Product[Factorial[3 i + 1]/Factorial[a + i], {i, 0, a - 1}]*
    Product[Factorial[3 i + 1]/Factorial[n - a + i], {i, 0, n - a - 1}], {a, 1, n - 1}], {n, 1, 13}]

a(n) = 2*Product_{i=0..n-1} (3i+1)!/(n+i)!+ Sum_{a=1..n-1}(Product_{i=0..a-1} (3i+1)!/(a+i)!)*(Product_{i=0..n-a-1} (3i+1)!/(n-a+i)!).

a(n) ~ exp(1/36) * Pi^(1/3) * 3^(3*n^2/2 - 7/36) / (A^(1/3) * Gamma(1/3)^(2/3) * n^(5/36) * 2^(2*n^2 - 17/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 18 2023

A383875 Number of pairs in the Bruhat order of type A_n.

Original entry on oeis.org

1, 3, 19, 213, 3781, 98407, 3550919

Offset: 0

Dmitry I. Ignatov, May 18 2025

The number of ordered pairs in the Bruhat poset of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).

			For n=0, the only element is 1 (identity) so a(0)=1.
For n=1 the elements are 1 (identity) and s1. The order relation consists of pairs (1, 1), (1, s1), and (s1, s1). So a(1) = 3.
For n=2 the line (Hasse) diagram is below.
       s1*s2*s1
        /   \
      s2*s1 s1*s2
       |  X  |
       s2    s1
        \   /
          1
The order relation consists of the six reflexive pairs, the eight pairs shown in the diagram as edges, and the five pairs (1, s2*s1), (1, s1*s2), (1, s1*s2*s1), (s1, s1*s2*s1), and (s2, s1*s2*s1). So a(2) = 6+8+5 = 19.

A. Bjorner and 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), A002538 (edges in the cover relation), A005130 (the size of Dedekind-MacNeille completion), A384061 (antichains), A384062 (maximal antichains).

a(0)=1 prepended by Sara Billey, Jul 02 2025

A384687 Number of elements in the Dedekind-MacNeille completion of the Bruhat order on D_n.

Original entry on oeis.org

4, 42, 1292, 114976, 29735760

Offset: 2

Dmitry I. Ignatov, Jun 07 2025

This sequence is the number of elements in the Dedekind-MacNeille completion (completion by cuts) of the Bruhat order of the Weyl group D_n. It is a type D analog of A378072.

			For n=2 the Bruhat order on D_2 consists of four elements, 1 (identity), s1, s2, and s2*s1. Its completion forms the diamond lattice and coincides with the order.
  s2*s1
   / \
  s1 s2
   \ /
    1

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

Cf. A002866 (group D_n order), A005130 (completion for A_n), A378072 (completion for B_n).

A384959 Number of chains in the Bruhat order of type A_n.

Original entry on oeis.org

4, 36, 4524, 15166380, 2010484649524, 14206021962108887860

Offset: 1

Dmitry I. Ignatov, Jun 13 2025

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

A. Bjorner and 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.
A. Postnikov and R. P. Stanley, Chains in the Bruhat order, J. Algebr. Comb., 29 (2009), 133-174.

Cf. A061710 (maximal chains), A000142 (the order size), A005130 (the size of Dedekind-MacNeille completion), A384061 (number of antichains).

cp's OEIS Frontend

A180512 Triangle of the number of alternating sign matrices according to the number of -1's.

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A293932 Number of circularly chained n-tuples of 3 X 3 alternating sign matrices.

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A306397 Sum of the coefficients in the Schur expansion of Product_{1<=i<=j<=n} (1+x_i+x_j), which is the total Chern class for the vector bundle Sym^2(E) where E is a vector bundle over a smooth complex projective variety of rank n with Chern roots x_1,...,x_n.

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A342181 Product of first n Robbins numbers.

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A363685 Irregular triangle read by rows: T(n,k) is the number of descending plane partitions of order n with the sum of the parts equal to k.

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A363689 Number of alternating sign matrices of size n which are indecomposable by direct and skew sums.

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A366449 Number of smooth discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n} having neutral element/absorbing element.

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Programs

Mathematica

Formula

A383875 Number of pairs in the Bruhat order of type A_n.

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A384687 Number of elements in the Dedekind-MacNeille completion of the Bruhat order on D_n.

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A384959 Number of chains in the Bruhat order of type A_n.

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