A209612 -id:A209612 - cp's OEIS frontend

A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1

Offset: 1

Emeric Deutsch, Nov 24 2003

Rows 2, 4, 6, ... give A088459.
Diagonal sums are in A088518(n-1). - Philippe Deléham, Jan 04 2009
Row sums are in A001405(n). - Philippe Deléham, Jan 04 2009
Subtriangle (1 <= k <= n) of triangle T(n,k), 0 <= k <= n, read by rows, given by A101455 DELTA A056594 := [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Also, number of symmetric noncrossing partitions of an n-set with k blocks. - Andrew Howroyd, Nov 15 2017
From Roger Ford, Oct 17 2018: (Start)
T(n,k) = t(n+2,d) where t(n,d) is the number of different semi-meander arch depth listings with n top arches and with d the depth of the deepest embedded arch.
Examples:    /\         semi-meander with 5 top arches
            //\\   /\   2 arches are at depth=0 (no covering arches)
           ///\\\ //\\  2 arches are at depth=1 (1 covering arch)
      (0)(1)(2)         1 arch is at depth=2 (2 covering arches)
       2, 2, 1 is the listing for this t(5,2)
             /\      semi-meander with 5 top arches
            /  \     (0)(1)
     /\ /\ //\/\\     3, 2  is the listing for this t(5,1)
a(6,5) = t(8,5)= 3 {2,1,1,1,2,1; 2,1,2,1,1,1; 3,1,1,1,1,1} (End)

			Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,   1;
  1,  2,  4,   2,   1;
  1,  3,  6,   6,   3,    1;
  1,  3,  9,   9,   9,    3,    1;
  1,  4, 12,  18,  18,   12,    4,    1;
  1,  4, 16,  24,  36,   24,   16,    4,    1;
  1,  5, 20,  40,  60,   60,   40,   20,    5,    1;
  1,  5, 25,  50, 100,  100,  100,   50,   25,    5,    1;
  1,  6, 30,  75, 150,  200,  200,  150,   75,   30,    6,   1;
  1,  6, 36,  90, 225,  300,  400,  300,  225,   90,   36,   6,   1;
  1,  7, 42, 126, 315,  525,  700,  700,  525,  315,  126,  42,   7,  1;
  1,  7, 49, 147, 441,  735, 1225, 1225, 1225,  735,  441, 147,  49,  7, 1;
  1,  8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1;
  ...
a(6,2)=3 because we have UUUDDDUUUDDD, UUUUDDUUDDDD, UUUUUDUDDDDD, where
U=(1,1), D=(1,-1).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Hyunsoo Cho, JiSun Huh, and Jaebum Sohn, The (s, s + d, ..., s + pd)-core partitions and the rational Motzkin paths, arXiv:2001.06651 [math.CO], 2020.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.
Vladimir Shevelev, Several remarks on A088855, Seqfan thread, Nov 19 2017.

Cf. A005566, A018224, A056594, A084938, A088459, A101455, A209612, A247644.
Cf. A001405 (row sums), A088459, A088518 (diagonal sums).
Column 2 is A008619, column 3 is A002620, column 4 is A028724, column 5 is A028723, column 6 is A028725, column 7 is A331574.

[(&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 08 2022

T[n_, k_] := Binomial[Quotient[n-1, 2], Quotient[k-1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]];
Table[T[n, k], {n,13}, {k,n}]//Flatten (* Jean-François Alcover, Jun 07 2018 *)

T(n,k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ Andrew Howroyd, Nov 15 2017

def A088855(n,k): return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A088855(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 08 2022

T(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.
G.f.: 2*u/(u*v + sqrt(x*y*u*v)) - 1, where x = 1+z+t*z, y = 1+z-t*z, u = 1-z+t*z, v = 1-z-t*z.
Triangle T(n,k), 0 <= k <= n, given by A101455 DELTA A056594 begins: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,2,1; 0,1,2,4,2,1; 0,1,3,6,6,3,1; 0,1,3,9,9,9,3,1; ... - Philippe Deléham, Jan 03 2009
From G. C. Greubel, Apr 08 2022: (Start)
T(n, n-k+1) = T(n, k).
T(2*n-1, n) = A018224(n-1), n >= 1.
T(2*n, n) = A005566(n-1), n >= 1. (End)

Keyword:tabl added Philippe Deléham, Jan 25 2010

A111275 Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.

Original entry on oeis.org

1, 2, 3, 6, 10, 24, 49, 130, 336, 980, 2904, 9176, 29432, 97356, 326399, 1111770, 3825238, 13293456, 46553116, 164200028, 582706692, 2079517924, 7458493728, 26874412064, 97241528200, 353223728624, 1287668381250, 4709805627484

Offset: 1

David Callan and Len Smiley, Oct 21 2005

nonn

These may be viewed as bracelets (able to be turned over in space) designed with n beads on a circle, each of which is a vertex of exactly one of a set of non-touching internal polygons (which may be 1-gons (beads), 2-gons (2 connected beads), etc.).

S.-C. Chang, J. L. Jacobsen, J. Salas, R. Shrock, "Exact Potts model partition functions for strips of the triangular lattice", J. Statist. Phys. 114, nos.3-4, pp. 763-823 [Corollary 2.1]
Motzkin, T. "Relations Between Hypersurface Cross Ratios and a Combinatorial Formula for Partitions of a Polygon for Permanent Preponderance and for Non-Associative Products." Bull. Amer. Math. Soc. 54, page 360, 1948.

D. Callan and L. Smiley, Non-crossing Partitions under Rotation and Reflection
Tilman Piesk, Partition related number triangles
L. Smiley, a(6)
L. Smiley, a(5)

Cf. A209612.

Table[Length[EquivalenceClasses[NCPartitions[n], groupDihedral[n]]], {n, 9}]

(A054357(n) + A001405(n))/2.

A303929 Array read by antidiagonals: T(n,k) is the number of noncrossing partitions up to rotation and reflection composed of n blocks of size k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 6, 1, 1, 1, 1, 3, 8, 13, 12, 1, 1, 1, 1, 4, 11, 34, 49, 27, 1, 1, 1, 1, 4, 16, 60, 169, 201, 65, 1, 1, 1, 1, 5, 20, 109, 423, 1019, 940, 175, 1, 1, 1, 1, 5, 26, 167, 918, 3381, 6710, 4643, 490, 1

Offset: 0

Andrew Howroyd, May 02 2018

			=================================================================
n\k| 1   2    3     4      5       6       7       8        9
---+-------------------------------------------------------------
0  | 1   1    1     1      1       1       1       1        1 ...
1  | 1   1    1     1      1       1       1       1        1 ...
2  | 1   1    1     1      1       1       1       1        1 ...
3  | 1   2    2     3      3       4       4       5        5 ...
4  | 1   3    5     8     11      16      20      26       32 ...
5  | 1   6   13    34     60     109     167     257      359 ...
6  | 1  12   49   169    423     918    1741    3051     4969 ...
7  | 1  27  201  1019   3381    9088   20569   41769    77427 ...
8  | 1  65  940  6710  29335   96315  259431  607696  1280045 ...
9  | 1 175 4643 47104 266703 1072187 3417520 9240444 22066742 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Wikipedia, Noncrossing partition

Columns 2..5 are A006082(n+1), A082938, A303870, A303871.

Cf. A111275, A209612, A211359, A303694, A303875.

u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #]&] + DivisorSum[GCD[n - 1, k], EulerPhi[#]*Binomial[n*k/#, (n - 1)/#]&])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
Table[T[n - k, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 14 2018, translated from PARI *)

\\ here c(n,k) is A303694
u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
c(n, k)={if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(k*d, d)) + sumdiv(gcd(n-1, k), d, eulerphi(d)*binomial(n*k/d, (n-1)/d)))/(k*n) - binomial(k*n, n)/(n*(k-1)+1))}
T(n,k)={(1/2)*(c(n,k) + if(n==0, 1, if(k%2, if(n%2, 2*u(n\2,k,(k+1)/2), u(n/2,k,1) + u(n/2-1,k,k)), e(n,k) + if(n%2, u(n\2,k,k/2)))/2))}

A209805 Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 10, 10, 3, 1, 1, 3, 15, 25, 15, 3, 1, 1, 4, 26, 64, 64, 26, 4, 1, 1, 4, 38, 132, 196, 132, 38, 4, 1, 1, 5, 56, 256, 536, 536, 256, 56, 5, 1, 1, 5, 75, 450, 1260, 1764, 1260, 450, 75, 5, 1

Offset: 1

Tilman Piesk, Mar 13 2012

Like the Narayana triangle A001263 (and unlike A152175) this triangle is symmetric.
The diagonal entries are 1, 1, 4, 25, 196, 1764, ... which is probably sequence A001246 - the squares of the Catalan numbers.
The above conjecture about the diagonal entries T(2*n-1, n) is true since gcd(2*n-1, n) = gcd(2*n-1, n-1) = 1 and then T(2*n-1, n) simplifies to A001246(n-1) using the formula given below. - Andrew Howroyd, Nov 15 2017

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   2,   2,   1;
  1,   2,   4,   2,   1;
  1,   3,  10,  10,   3,   1;
  1,   3,  15,  25,  15,   3,   1;
  1,   4,  26,  64,  64,  26,   4,   1;
  1,   4,  38, 132, 196, 132,  38,   4,   1;
  1,   5,  56, 256, 536, 536, 256,  56,   5,   1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Tilman Piesk, Partition related number triangles (Wikiversity)
Tilman Piesk, Brute force MATLAB code used to calculate the rows (Pastebin)

Cf. A054357 (row sums), A001246 (square Catalan numbers).

Cf. A001263, A103371, A132812, A209612.

b[n_, k_] := Binomial[n-1, n-k] Binomial[n, n-k];
T[n_, k_] := (DivisorSum[GCD[n, k], EulerPhi[#] b[n/#, k/#]&] + DivisorSum[GCD[n, k - 1], EulerPhi[#] b[n/#, (n + 1 - k)/#]&] - k Binomial[n, k]^2/(n - k + 1))/n;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

b(n,k)=binomial(n-1,n-k)*binomial(n,n-k);
T(n,k)=(sumdiv(gcd(n,k), d, eulerphi(d)*b(n/d,k/d)) + sumdiv(gcd(n,k-1), d, eulerphi(d)*b(n/d,(n+1-k)/d)) - k*binomial(n,k)^2/(n-k+1))/n; \\ Andrew Howroyd, Nov 15 2017

T(n,k) = (1/n)*((Sum_{d|gcd(n,k)} phi(d)*A103371(n/d-1,k/d-1)) + (Sum_{d|gcd(n,k-1)} phi(d)*A103371(n/d-1,(n+1-k)/d-1)) - A132812(n,k)). - Andrew Howroyd, Nov 15 2017

A211359 Triangle read by rows: T(n,k) is the number of noncrossing partitions up to rotation and reflection of an n-set that contain k singleton blocks.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 2, 3, 2, 2, 0, 1, 5, 4, 8, 3, 3, 0, 1, 6, 11, 12, 12, 4, 3, 0, 1, 14, 21, 39, 24, 22, 5, 4, 0, 1, 22, 55, 84, 85, 48, 30, 7, 4, 0, 1, 51, 124, 245, 228, 190, 82, 46, 8, 5, 0, 1, 95, 327, 620, 730, 570, 350, 136, 60

Offset: 0

Tilman Piesk, Apr 12 2012

			From _Andrew Howroyd_, May 02 2018: (Start)
Triangle begins:
   1;
   0,   1;
   1,   0,   1;
   1,   1,   0,   1;
   2,   1,   2,   0,   1;
   2,   3,   2,   2,   0,  1;
   5,   4,   8,   3,   3,  0,  1;
   6,  11,  12,  12,   4,  3,  0, 1;
  14,  21,  39,  24,  22,  5,  4, 0, 1;
  22,  55,  84,  85,  48, 30,  7, 4, 0, 1;
  51, 124, 245, 228, 190, 82, 46, 8, 5, 0, 1;
  ...
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..90 from Tilman Piesk)

Column k=0 is A303931.
Row sums are A111275.
Cf. A209612, A211357, A303875.

\\ See A303875 for NCPartitionsModDihedral
{ my(rows=Vec(NCPartitionsModDihedral(vector(10, k, if(k==1,y,1)))));
for(n=1, #rows, for(k=0, n-1, print1(polcoeff(rows[n], k), ", ")); print; ) } \\ Andrew Howroyd, May 02 2018

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A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

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A111275 Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.

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A303929 Array read by antidiagonals: T(n,k) is the number of noncrossing partitions up to rotation and reflection composed of n blocks of size k.

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A209805 Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations.

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Mathematica

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A211359 Triangle read by rows: T(n,k) is the number of noncrossing partitions up to rotation and reflection of an n-set that contain k singleton blocks.

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PARI