Arvind Ayyer - cp's OEIS frontend

A373625 Sum of all entries in character table of the hyperoctahedral group B_n.

1, 2, 8, 26, 112, 410, 1860, 8074, 40376, 199050, 1085232, 5923394, 34842408, 206403234, 1295653484, 8219293954, 54613967584, 367414298386, 2567777927672, 18187100499306, 133016727225888, 986352813933034, 7518613974827732, 58110359176236314, 460095738657984024

Offset: 0

Arvind Ayyer, Jun 11 2024

nonn

			a(2) = 8 because the character table of B_2 is  [[1  1  1  1  1], [ 1 -1 -1  1  1], [ 1 -1  1 -1  1], [ 1  1 -1 -1  1], [ 2  0  0  0 -2]].

Andrew Howroyd, Table of n, a(n) for n = 0..500
Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036, [math.RT], 2024.

Cf. A001147, A376826, A082733.

\\ here B(k,n) is o.g.f. of column k of A376826.
B(k,n)={serlaplace(exp(2*x + k*x^2/2 + O(x*x^n)))}
seq(n)={my(d=serlaplace(1/sqrt(1 - 2*x + O(x*x^(n\2))))); Vec(prod(i=1, (n+1)\2, subst(d + O(x^(n\(2*i)+1)), x, 2*i*x^(2*i))^(2-i%2) * subst(B(4*i-2, n\(2*i-1)), x, x^(2*i-1))))} \\ Andrew Howroyd, Oct 07 2024

G.f.: Product_{i >= 1} D(4*i*x^(4*i)) * D(2*i*x^(2*i)) * R(2*i-1, x^(2*i-1)), where D(x) is the g.f. of A001147, R(r, x) = Sum_{k>=0} c(r,k)*x^k and c(r,n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * 2^(n-k) * (2*k-1)!! * r^k. [edited by Andrew Howroyd, Oct 07 2024]

G.f.: Product_{i >= 1} D(4*i*x^(4*i)) * D(2*i*x^(2*i)) * B(4*i-2, x^(2*i-1)), where D(x) is the g.f. of A001147 and B(k,x) is the g.f. of column k of A376826. - Andrew Howroyd, Oct 07 2024

a(0)=1 prepended and a(10) onwards from Andrew Howroyd, Oct 06 2024

A181119 Number of transpose-complementary plane partitions of n.

Original entry on oeis.org

1, 2, 84, 81796, 1844536720, 962310111888300, 11608208114358751650000, 3236574482779383546336417240000, 20853456581643133066208521560263633137920, 3104385823530881109001458753652585998600603921849920, 10676554307318599842868990948461304923921623250562199975300214736

Offset: 0

Arvind Ayyer, Jan 21 2011

The complement of a plane partition inside an m X m X m cube consists of the boxes which are within the cube, but not in the plane partition, rotated in an appropriate way.

a(n) is the number of plane partitions inside an 2n X 2n X 2n cube whose (matrix) transpose when written as an 2n X 2n array is the same as its complement.

			When n=2, there are two transpose-complementary plane partitions,
[1 1] and [2 1], both of whose transpose and complement is equal to themselves.
[1 1]     [1 0]

R. P. Stanley, Symmetries of Plane Partitions, J. Comb. Theory Ser. A 43 (1986), 103-113.
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
Wikipedia, Plane partition

Cf. A008793, A051255, A078920, A123352.

Table[Binomial[3n-1,n]Product[(2n+i+j+1)/(i+j+1),{i,1,2n-2}, {j,i,2n-2}], {n,0,10}] (* Harvey P. Dale, Jan 27 2012 *)

a(n) = binomial(3*n-1,n)*prod(i=1,2*n-2,prod(j=i,2*n-2,(2*n+i+j+1)/(i+j+1))); \\ Michel Marcus, Jun 18 2015

a(n) = binomial(3n-1,n)*Product(i=1..2n-2,Product(j=i..2n-2,(2n+i+j+1)/(i+j+1))).

a(n) ~ exp(1/24) * 3^(9*n^2 - 3*n/2 - 1/24) / (sqrt(A) * n^(1/24) * 2^(12*n^2 - n - 1/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 28 2015

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

Original entry on oeis.org

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

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

A157513 Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.

Original entry on oeis.org

1, 1, 1, 2, 7, 2, 5, 37, 38, 5, 14, 177, 390, 187, 14, 42, 806, 3065, 3175, 874, 42, 132, 3566, 20742, 37260, 22254, 3958, 132, 429, 15485, 127575, 351821, 365433, 141442, 17548, 429, 1430, 66373, 734332, 2876886, 4597444, 3100670, 839068, 76627, 1430

Offset: 0

Arvind Ayyer, Mar 02 2009

The first and the last terms in each row are Catalan numbers. The sum in each row gives the Gessel sequence.

			For n=2, there are 2 walks of length 4 where the diagonal steps (1,1) and (-1,-1) occur zero times [(1,0),(1,0),(-1,0),(-1,0)] and [(1,0),(-1,0),(1,0),(-1,0)];
7 walks where the diagonal steps occur once [(1,0),(-1,0),(1,1),(-1,-1)], [(1,1),(-1,-1),(1,0),(-1,0)],  [(1,0),(1,1),(-1,0),(-1,-1)],  [(1,0),(1,1),(-1,-1),(-1,0)],  [(1,1),(1,0),(-1,0),(-1,-1)],  [(1,1),(1,0),(-1,-1),(-1,0)],  [(1,1),(-1,0),(1,0),(-1,-1)];
and finally 2 walks where the diagonal steps occur twice [(1,1),(1,1),(-1,-1),(-1,-1)] and [(1,1),(-1,-1),(1,1),(-1,-1)].
Triangle begins:
1;
1,     1;
2,     7,    2;
5,    37,   38,    5;
14,  177,  390,  187,   14;
42,  806, 3065, 3175,  874,  42;

Alois P. Heinz, Table of n, a(n) for n = 0..5049
Arvind Ayyer, Towards a human proof of Gessel's conjecture, arXiv:0902.2329 [math.CO], 2009.
Manuel Kauers, Christoph Koutschan and Doron Zeilberger, Proof of Ira Gessel's Lattice Path Conjecture
Marko Petkovsek and Herbert S. Wilf, On a conjecture of Ira Gessel, arXiv:0807.3202 [math.CO], 2008.

Cf. A135404, A000531, A000108.

b:= proc(n, k, t, x, y) option remember; `if` (min(n, x, y, k, t, n-x)<0, 0, `if` (n=0, `if` (max(n, k, t)=0, 1, 0), b(n-1, k-1, t, x+1, y+1) +b(n-1, k, t, x+1, y) +b(n-1, k, t, x-1, y) +b(n-1, k, t-1, x-1, y-1))) end: T:= (n,k)-> b(2*n, k, k, 0, 0):
seq (seq (T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Jul 04 2011

b[n_, k_, t_, x_, y_] := b[n, k, t, x, y] = If[Min[n, x, y, k, t, n-x] < 0, 0, If[n == 0, If[Max[n, k, t] == 0, 1, 0], b[n-1, k-1, t, x+1, y+1] + b[n - 1, k, t, x+1, y] + b[n-1, k, t, x-1, y] + b[n-1, k, t-1, x-1, y-1]]]; T[n_, k_] := b[2*n, k, k, 0, 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)

A135452 Number of different multisets of differences between ends of n non-intersecting chords joining 2n labeled points around a circle.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 33, 71, 117, 242, 421, 877, 1468

Offset: 1

Arvind Ayyer, Dec 14 2007, definition corrected Jan 04 2007, Jan 07 2007

Represent a set of chords as a collection of pairs of integers. For example, if n=3, one possible connectivity is {{1,4},{2,3},{5,6}}.
Define the D-set of a connectivity to be the multiset of differences between connected pairs. In the above example the D-set is {1,1,3}. Since the numbers are on a circle, we can take two possible differences. We take the smaller of the two. Hence the maximal difference can be at most n or n-1 depending on whether n is odd or even. Is another example: the D-set of {{1,6},{2,3},{4,5}} is {1,1,1}.
Then the sequence gives the number of distinct D-sets of all possible connectivities.
While it is true that if two connectivities have different D-sets they are inequivalent, the converse is not true. consider n=6: Both {{4, 5}, {6, 11}, {2, 3}, {8, 9}, {7, 10}, {1, 12}} and {{4, 5}, {1, 6}, {2, 3}, {8, 9}, {7, 10}, {11, 12}} have the same D-set, namely {1,1,1,1,3,5} but they are inequivalent.

Cf. A001008, A054357.

A133107 Number of Ferrers diagrams with a single strictly smaller Ferrers puncture with the same orientation removed from the top with half-perimeter = n.

Original entry on oeis.org

1, 7, 32, 121, 410, 1294, 3888, 11273, 31826, 88041, 239734, 644758, 1717191, 4538129, 11919760, 31156313, 81125827, 210604604, 545462798, 1410226551, 3641097828, 9391872711, 24208902420, 62373915102, 160663604377

Offset: 6

Arvind Ayyer, Sep 11 2007

			The sequence starts with n=6 because the smallest such object whose illustration is below has a perimeter of 12. (1 denotes cell inside the Ferrers diagram.)
1 1
111

Arvind Ayyer, Sep 11 2007, Table of n, a(n) for n = 6..76

Cf. A057410, A057406, A133106.

G.f.: x^2*(-1 + 3*x - x^2 + (5*x^4 - 6*x^3 + 11*x^2 - 6*x + 1 + 4*x^6 - 12*x^5)^(1/2))/(2*(x^2 - 3*x + 1)*(1-2*x)^2)

A133106 Number of Ferrers diagrams with a single Ferrers puncture with the same orientation inscribed strictly inside with half-perimeter = n.

Original entry on oeis.org

1, 8, 41, 168, 602, 1968, 6021, 17512, 48950, 132496, 349258, 900368, 2277556, 5667936, 13906221, 33695208, 80746846, 191601872, 450642654, 1051472048, 2435679852, 5605044640, 12820922530, 29164511376, 66004709148, 148678206880

Offset: 8

Arvind Ayyer, Sep 11 2007

			The sequence starts with n=8 because the smallest such object whose cartoon is below has a perimeter of 16. (1 denotes cell outside the puncture and 2 denotes cell inside the puncture).
111
121
111

Charles R Greathouse IV, Table of n, a(n) for n = 8..3296

Cf. A057410, A057406, A133107.

Drop[CoefficientList[Series[(1-(1-4x^2)^(1/2))x^6/(2(2x-1)^4),{x,0,40}],x],8] (* Harvey P. Dale, Sep 21 2024 *)

a(n) = [(2*n^2-16*n+6)*a(n-1)+(4*n^2-68*n+240)*a(n-2)-(8*n^2-88*n+240)*a(n-3)]/(n^2-14*n+48) with a(6)=0, a(7)=0, a(8)=1.

G.f.: (1-(1-4*x^2)^(1/2))*x^6/(2*(2*x-1)^4).

A127618 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 4 with the steps (1,0), (0, 1), (2,0) and (0,2).

Original entry on oeis.org

1, 1, 5, 22, 117, 590, 3018, 15378, 78440, 399992, 2039852, 10402480, 53049048, 270531368, 1379614800, 7035549312, 35878823312, 182969359520, 933079279328, 4758375627808, 24266039468160, 123748253080832, 631072497876672

Offset: 0

Arvind Ayyer, Jan 20 2007

			a(2)=5 because we can reach (2,2) in the following ways:
(0,0),(1,0),(1,1),(2,1),(2,2)
(0,0),(2,0),(2,2)
(0,0),(1,0),(2,0),(2,2)
(0,0),(2,0),(2,1),(2,2)
(0,0),(1,0),(2,0),(2,1),(2,2)

Arvind Ayyer and Doron Zeilberger, The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points, arXiv:math/0610734 [math.CO], 2006-2007.
Index entries for linear recurrences with constant coefficients, signature (4,6,-2).

Cf. A000108, A046717, A122951, A127617, A127619, A127620.

Join[{1, 1}, LinearRecurrence[{4, 6, -2}, {5, 22, 117}, 21]] (* Jean-François Alcover, Dec 10 2018 *)
b[n_, k_] := Boole[n >= 0 && k >= 0 && 0 <= n-k <= 4];
T[0, 0] = T[1, 1] = 1; T[n_, k_] /; b[n, k] == 1 := T[n, k] = b[n-2, k]* T[n-2, k] + b[n-1, k]*T[n-1, k] + b[n, k-2]*T[n, k-2] + b[n, k-1]*T[n, k-1]; T[, ] = 0;
a[n_] := T[n, n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 03 2019 *)

G.f.: (1-3x-5x^2-2x^3+x^4)/(1-4x-6x^2+2x^3).

A127619 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 5 with the steps (1,0), (0, 1), (2,0) and (0,2).

Original entry on oeis.org

1, 1, 5, 22, 117, 654, 3674, 20763, 117349, 663529, 3751874, 21215245, 119963514, 678345474, 3835772387, 21689760681, 122646936325, 693519457822, 3921575652821, 22174944672838, 125390459051898, 709032985366923

Offset: 0

Arvind Ayyer, Jan 20 2007

			a(2)=5 because we can reach (2,2) in the following ways:
(0,0),(1,0),(1,1),(2,1),(2,2)
(0,0),(2,0),(2,2)
(0,0),(1,0),(2,0),(2,2)
(0,0),(2,0),(2,1),(2,2)
(0,0),(1,0),(2,0),(2,1),(2,2)

Arvind Ayyer and Doron Zeilberger, The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points, arXiv:math/0610734 [math.CO], 2006-2007.
Index entries for linear recurrences with constant coefficients, signature (5, 6, -11, -12, 4).

Cf. A000108, A046717, A122951, A127617, A127618, A127620.

LinearRecurrence[{5, 6, -11, -12, 4}, {1, 1, 5, 22, 117}, 22] (* Jean-François Alcover, Dec 10 2018 *)
b[n_, k_] := Boole[n >= 0 && k >= 0 && 0 <= n - k <= 5];
T[0, 0] = T[1, 1] = 1; T[n_, k_] /; b[n, k] == 1 := T[n, k] = b[n-2, k]* T[n-2, k] + b[n-1, k]*T[n-1, k] + b[n, k-2]*T[n, k-2] + b[n, k-1]*T[n, k-1]; T[, ] = 0;
a[n_] := T[n, n];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 03 2019 *)

G.f.: (1-4x-6x^2+2x^3)/(1-5x-6x^2+11x^3+12x^4-4x^5). [Typo corrected by Jean-François Alcover, Dec 10 2018]

cp's OEIS Frontend

User: Arvind Ayyer

A373625 Sum of all entries in character table of the hyperoctahedral group B_n.

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PARI

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A181119 Number of transpose-complementary plane partitions of n.

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Mathematica

PARI

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

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A180349 Gog words avoiding the subpattern 312.

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A157513 Triangle of numbers of walks in the quarter-plane, of length 2n beginning and ending at the origin using steps {(1,1), (1,0), (-1,0), (-1,-1)} (Gessel steps) arranged according to the number of times the steps (1,1) and (-1,-1) occur.

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Maple

Mathematica

A135452 Number of different multisets of differences between ends of n non-intersecting chords joining 2n labeled points around a circle.

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A133107 Number of Ferrers diagrams with a single strictly smaller Ferrers puncture with the same orientation removed from the top with half-perimeter = n.

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Keywords

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Formula

A133106 Number of Ferrers diagrams with a single Ferrers puncture with the same orientation inscribed strictly inside with half-perimeter = n.

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Keywords

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Programs

Mathematica

Formula

A127618 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 4 with the steps (1,0), (0, 1), (2,0) and (0,2).

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