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A256672 Number of idempotents in the Motzkin monoid of degree n.

1, 2, 7, 31, 153, 834, 4839, 29612, 188695, 1243746, 8428597, 58476481, 413893789, 2980489256, 21787216989, 161374041945, 1209258743839, 9155914963702, 69969663242487, 539189056700627

Offset: 0

Nick Loughlin, Apr 07 2015

a(n) is the number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle, such that when gluing the second half of one copy to the first half of the other so that each point k along the intersection is glued to n+1-k, the result is homotopic to the original.
a(n+1) > a(n) for every n.
The structure of the Motzkin monoid (and particularly its idempotents and some associated orderings) is governed intimately by the combinatorics of so-called Motzkin paths and Motzkin words, which are related to Dyck paths and words respectively by insertion of punctuation into the words, or marking/coloring subpaths.
Bounded above by A026945, strictly for n > 1. Bounded below by the square of A001006, strictly for n > 1.

			There is one empty graph, which is idempotent under the composition, hence a(0)=1.
There are two on 1 pair of points, the clique and the discrete graph; both are idempotents under the composition, hence a(1)=2.

I. Dolinka, J. East et al, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv: 1507.04838 [math.CO], 2015, Table 2.
Tom Halverson, Gelfand Models for Diagram Algebras, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013).
Tom Halverson and Mike Reeks, Gelfand Models for Diagram Algebras, Journal of Algebraic Combinatorics (2014)41, 229-255.
J. D. Mitchell et al., Semigroups - GAP package, Version 2.7.4, March, 2016.
J. D. Mitchell, Counting idempotents in a monoid of partitions, C++ program, October, 2016
Eliezer Posner, Kris Hatch, and Megan Ly, Presentation of the Motzkin Monoid, arXiv:1301.4518 [math.RT], 2013.

Cf. A001006, A026945.

a(9)-a(13) corrected and a(14)-a(16) computed using the Semigroups package for GAP added by James Mitchell, Apr 12 2016

a(17)-a(19) added by James Mitchell, Apr 01 2017

A256492 Number of idempotents in the partial Jones monoid.

Original entry on oeis.org

1, 2, 7, 24, 103, 416, 1998, 8822, 45661, 213674, 1167797, 5694690, 32445914, 163151262, 960580840, 4945645808, 29899013071, 156834641076, 968947169139

Offset: 0

Nick Loughlin, Mar 30 2015

The partial Jones monoid contains all the elements of the Motzkin monoid whose pictorial representatives are subgraphs of those in the Jones monoid. The number a(n) counts the idempotent elements in this monoid in each degree n, starting from zero. This monoid was discovered by the sequence's original author and a collaborator during work on a paper yet to appear at the time of posting.

			In degree at most 1, the idempotents are all partial identities, giving a(0)=1 and a(1)=2. In degree 2 ,there are 7; the four partial identities, the Temperly-Lieb cup-and-cap, and its 3 subpictures (one of which is the empty picture, which is also a partial identity, hence the overcount by 1).

V. F. R. Jones, The Potts model and the symmetric group, in: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993), World Sci. Publishing, 1994, 259-267.

Egri-Nagy Attila, Organic semigroup theory: ferns growing in the Jones/Temperley-Lieb monoid, on Computational Semigroup Theory at Wordpress, September 1, 2014.
I. Dolinka, J. East et al, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv: 1507.04838 [math.CO] (2015).
J. East, Egri-Nagy A., A. R. Francis, J. D. Mitchell, Finite Diagram Semigroups: Extending the Computational Horizon, arXiv:1502.07150 [math.GR], 2015.
K. Hatch, E. Ly, E. Posner, Presentation of the Motzkin Monoid, arXiv:1301.4518 [math.RT], 2013.
K. W. Lau & D. G. FitzGerald, Ideal Structure of the Kauffman and Related Monoids, Communications in Algebra, 30:7 (2006), 2617-2629. doi:10.1080/00927870600651414
J. D. Mitchell et al., Semigroups package for GAP.

a(11)-a(18) computed using the GAP package Semigroups and added by James Mitchell, May 21 2016

A241907 a(n) = floor( Catalan(2*n) / Catalan(n)^2 ).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 14, 17, 20, 23, 26, 29, 33, 36, 40, 44, 48, 52, 56, 60, 65, 69, 74, 78, 83, 88, 93, 98, 103, 108, 114, 119, 124, 130, 136, 141, 147, 153, 159, 165, 171, 177, 183, 189, 196, 202, 209, 215, 222, 229, 235, 242, 249, 256, 263, 270, 277, 284, 292, 299, 306, 314, 321, 329, 336, 344, 352, 360, 367, 375, 383, 391, 399, 408, 416, 424, 432, 441, 449, 457, 466, 474, 483, 492, 500, 509, 518, 527, 536, 545, 554, 563, 572, 581, 590, 599, 609, 618, 627, 637, 646, 656, 665, 675, 685, 694, 704, 714, 724, 734, 744, 753, 764, 774, 784, 794, 804, 814, 825, 835, 845, 856, 866, 877, 887, 898, 908, 919, 930, 940, 951, 962, 973, 984, 995, 1006

Offset: 0

Nick Loughlin, May 01 2014

This sequence is (roughly) the relative size of the Jones monoid J_n to its minimal ideal. Equivalently, this is roughly the reciprocal of the proportion of Dyck words of length 4n which can be factorized into two Dyck words, each of length 2n.

Wikipedia, Dyck word

Cf. A000108.

Digits:=200:
C:=n->binomial(2*n,n)/(n+1); f:=n->floor(C(2*n)/C(n)^2);[seq(f(n),n=0..100)]; # N. J. A. Sloane, May 21 2014

Table[Floor[CatalanNumber[2n]/CatalanNumber[n]^2],{n,0,140}] (* Harvey P. Dale, Oct 04 2015 *)

a(n) = floor( Catalan( 2*n ) / Catalan(n)^2 ).

Corrected by Harvey P. Dale, Oct 04 2015

A241906 a(n) = floor(bell(2n)/bell(n)^2), bell = A000110.

Original entry on oeis.org

1, 2, 3, 8, 18, 42, 102, 248, 611, 1525, 3845, 9787, 25118, 64944, 169047, 442727, 1165990, 3086692, 8210400, 21936230, 58851484, 158502600, 428446818, 1162110731, 3162318827, 8631705612, 23629386708, 64865101678, 178531867765, 492622401009, 1362567996602, 3777490059587, 10495626146222, 29223682273897, 81535625627546, 227935763726546, 638409001899851

Offset: 0

Nick Loughlin, May 01 2014

nonn

a(n) is the largest integer smaller than the (reciprocal) proportion of partitions of the set {1,..,2n} that refine the partition {1,..,n|n+1,..,2*n}.

Cf. A000110.

GAP
```
QuoInt(Bell(2*n),Bell(n)^2)
```

Table[Floor[BellB[2*n]/BellB[n]^2], {n,0,30}] (* Vaclav Kotesovec, Jul 23 2021 *)

A199044 The number of identity elements of length n in Z*Z^2.

Original entry on oeis.org

1, 0, 6, 0, 74, 0, 1140, 0, 19562, 0, 357756, 0, 6824684, 0, 134166696, 0, 2697855082, 0, 55213424556, 0, 1146078241284, 0, 24067465856088, 0, 510351502965548, 0, 10911807871502232, 0, 234970037988773560, 0, 5091149074269149520, 0, 110912377099411850090, 0

Offset: 0

Nick Loughlin, Nov 02 2011

Z*Z^2 is the free product of the free group on one letter (say, x) and the free abelian group on two letters (say, y and z).
Viewed as the quotient of the free group F on three letters {x,y,z} by the normal subgroup generated by the commutator [y,z], the sequence gives the number of words in F of length n that are sent to the identity in Z*Z^2 under the quotient map.
Note that odd-numbered terms are zero.

			The identity from the free group F maps to the identity in Z*Z^2, and is the only word of length zero in F, so a(0)=1.
The group Z*Z^2 maps onto the direct product C_2^3, the group of exponent 2 with 8 elements. Therefore no elements of odd length are sent to the identity and thus a(2i-1)=0 for all positive integers i.
The only word of length zero is the empty word, which vacuously represents the identity. Therefore, a_0=1.
For n=2, there are a_2=6 identities; each is a (positive or negative) generator x,y, or z, followed or preceded by its inverse. We have the words x*x^-1, y*y^-1, z*z^-1, plus the reverse of each.

Derek F. Holt, Sarah Rees, Claas E. Röver, and Richard M. Thomas, Groups with Context-Free Co-Word Problem, J. London Math. Soc. (2005) 71 (3): 643-657. doi: 10.1112/S002461070500654X
Brough, Tara Rose, Groups with poly-context-free word problem, PhD thesis (2010), University of Warwick.

Nick Loughlin, Table of n, a(n) for n = 0..881

Edited by Max Alekseyev, Jan 24 2012

Edited by Nick Loughlin, Mar 12 2012

A152595 a(n) = the number of integers i in [1,n] that can be expressed as the sum of two squares of positive integers.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 24, 25, 25, 25, 25, 25, 25, 25

Offset: 1

Nick Loughlin, Dec 09 2008

# Use se(n) to find the first n terms of the sequence and te(n) to find the n-th term. se:=proc(n) local L,S,k,m,i,j: L:=[]:S:={}: for k from 1 to n do m:=ceil(sqrt(k/2)): for i from 1 to m do for j from 1 to m do if i^2+j^2<=k then S:={op(S),i^2+j^2}: fi:od:od: L:=[op(L),nops(S)]: od: end proc: te:=proc(n) local S,m,i,j: m:=ceil(sqrt(n/2)): S:={}: for i from 1 to m do for j from 1 to m do if i^2+j^2<=n then S:={op(S),i^2+j^2}: fi:od:od: return nops(S): end proc:

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A256672 Number of idempotents in the Motzkin monoid of degree n.

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A256492 Number of idempotents in the partial Jones monoid.

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A241907 a(n) = floor( Catalan(2*n) / Catalan(n)^2 ).

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A241906 a(n) = floor(bell(2n)/bell(n)^2), bell = A000110.

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A199044 The number of identity elements of length n in Z*Z^2.

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Author

Keywords

Comments

Examples

References

Links

Extensions

A152595 a(n) = the number of integers i in [1,n] that can be expressed as the sum of two squares of positive integers.

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Author

Keywords

Programs

Maple