Skip to main content

Fuss–Catalan number









Fuss–Catalan number


From Wikipedia, the free encyclopedia

Jump to navigation
Jump to search






In combinatorial mathematics and statistics, the Fuss–Catalan numbers are numbers of the form


Am(p,r)≡rmp+r(mp+rm)=rm!∏i=1m−1(mp+r−i)=rΓ(mp+r)Γ(1+m)Γ(m(p−1)+r+1).{displaystyle A_{m}(p,r)equiv {frac {r}{mp+r}}{binom {mp+r}{m}}={frac {r}{m!}}prod _{i=1}^{m-1}(mp+r-i)=r{frac {Gamma (mp+r)}{Gamma (1+m)Gamma (m(p-1)+r+1)}}.}{displaystyle A_{m}(p,r)equiv {frac {r}{mp+r}}{binom {mp+r}{m}}={frac {r}{m!}}prod _{i=1}^{m-1}(mp+r-i)=r{frac {Gamma (mp+r)}{Gamma (1+m)Gamma (m(p-1)+r+1)}}.}

They are named after N. I. Fuss and Eugène Charles Catalan.


In some publications this equation is sometimes referred to as Two-parameter Fuss–Catalan numbers or Raney numbers. The implication is the single-parameter Fuss-Catalan numbers are when r=1 and p=2.




Contents






  • 1 Uses


  • 2 Special Cases


  • 3 Examples


  • 4 Algebra


    • 4.1 Recurrence


    • 4.2 Generating Function


    • 4.3 Recursion Representation


    • 4.4 Alternate Representations




  • 5 See also


  • 6 References





Uses[edit]


The Fuss-Catalan represents the number of legal permutations or allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the Binomial Coefficient. The key difference between Fuss-Catalan and the Binomial Coefficient is that there are no "illegal" arrangement permutations within Binomial Coefficient, but there are within Fuss-Catalan. An examples of legal and illegal permutations can be better demonstrated by a specific problem such as balanced brackets (see Dyck language).


A general problem is to count the number of balanced brackets (or legal permutations) that a string of m open and m closed brackets forms (total of 2m brackets). By legally arranged, the following rules apply:



  • For the sequence as a whole, the number of open brackets must equal the number of closed brackets

  • Working along the sequence, the number of open brackets must be greater than the number of closed brackets


As an numeric example how many combinations can 3 pairs of brackets be legally arranged? From the Binomial interpretation there are (2mm){displaystyle {tbinom {2m}{m}}}{displaystyle {tbinom {2m}{m}}} or numerically (63){displaystyle {tbinom {6}{3}}}{displaystyle {tbinom {6}{3}}} = 20 ways of arranging 3 open and 3 closed brackets. However, there are fewer legal combinations than these when all of the above restrictions apply. Evaluating these by hand, there are 5 legal combinations, namely: ()()(); (())(); ()(()); (()()); ((())). This corresponds to the Fuss-Catalan formula when p=2, r=1 which is the Catalan number formula 12m+1(2mm){displaystyle {tfrac {1}{2m+1}}{tbinom {2m}{m}}}{displaystyle {tfrac {1}{2m+1}}{tbinom {2m}{m}}} or 14(63){displaystyle {tfrac {1}{4}}{tbinom {6}{3}}}{displaystyle {tfrac {1}{4}}{tbinom {6}{3}}}=5. By simple subtraction, there are mm+1(2mm){displaystyle {tfrac {m}{m+1}}{tbinom {2m}{m}}}{displaystyle {tfrac {m}{m+1}}{tbinom {2m}{m}}} or 34(63){displaystyle {tfrac {3}{4}}{tbinom {6}{3}}}{displaystyle {tfrac {3}{4}}{tbinom {6}{3}}} =15 illegal combinations. To further illustrate the subtlety of the problem, if one were to persist with solving the problem just using the Binomial formula, it would be realised that the 2 rules imply that the sequence must start with an open bracket and finish with a closed bracket. This implies that there are (2m−2m−1){displaystyle {tbinom {2m-2}{m-1}}}{displaystyle {tbinom {2m-2}{m-1}}} or (42){displaystyle {tbinom {4}{2}}}{displaystyle {tbinom {4}{2}}}=6 combinations. This is inconsistent with the above answer of 5, and the missing combination is: ())((), which is illegal and would complete the binomial interpretation.


Whilst the above is a concrete example Catalan numbers, similar problems can be evaluated using Fuss-Catalan formula:




  • Computer Stack: ways of arranging and completing a computer stack of instructions, each time step 1 instruction is processed and p new instructions arrive randomly. If at the beginning of the sequence there are r instructions outstanding.


  • Betting: ways of losing all money when betting. A player has a total stake pot that allows them to make r bets, and plays a game of chance that pays p times the bet stake.


  • Tries: Calculating the number of order m tries on n nodes.[1]



Special Cases[edit]




A0(p,r)=1{displaystyle A_{0}(p,r)=1}{displaystyle A_{0}(p,r)=1};


A1(p,r)=r{displaystyle A_{1}(p,r)=r}{displaystyle A_{1}(p,r)=r};


A2(p,1)=p{displaystyle A_{2}(p,1)=p}{displaystyle A_{2}(p,1)=p}.


If p=0{displaystyle p=0}p=0, we recover the Binomial coefficients Am(0,r)=(rm){displaystyle A_{m}(0,r){=}{binom {r}{m}}}{displaystyle A_{m}(0,r){=}{binom {r}{m}}}




Am(0,1)=1,1{displaystyle A_{m}(0,1)=1,1}{displaystyle A_{m}(0,1)=1,1};


Am(0,2)=1,2,1{displaystyle A_{m}(0,2)=1,2,1}{displaystyle A_{m}(0,2)=1,2,1};


Am(0,3)=1,3,3,1{displaystyle A_{m}(0,3)=1,3,3,1}{displaystyle A_{m}(0,3)=1,3,3,1};


Am(0,4)=1,4,6,4,1{displaystyle A_{m}(0,4)=1,4,6,4,1}{displaystyle A_{m}(0,4)=1,4,6,4,1}.


If p=1{displaystyle p=1}p=1, Pascal's Triangle appears, read along diagonals:




Am(1,1)=1,1,1,1,1,1,1,1,1,1,…{displaystyle A_{m}(1,1)=1,1,1,1,1,1,1,1,1,1,ldots }{displaystyle A_{m}(1,1)=1,1,1,1,1,1,1,1,1,1,ldots };


Am(1,2)=1,2,3,4,5,6,7,8,9,10,…{displaystyle A_{m}(1,2)=1,2,3,4,5,6,7,8,9,10,ldots }{displaystyle A_{m}(1,2)=1,2,3,4,5,6,7,8,9,10,ldots };


Am(1,3)=1,3,6,10,15,21,28,35,45,55,…{displaystyle A_{m}(1,3)=1,3,6,10,15,21,28,35,45,55,ldots }{displaystyle A_{m}(1,3)=1,3,6,10,15,21,28,35,45,55,ldots };


Am(1,4)=1,4,10,20,35,56,84,120,165,220,…{displaystyle A_{m}(1,4)=1,4,10,20,35,56,84,120,165,220,ldots }{displaystyle A_{m}(1,4)=1,4,10,20,35,56,84,120,165,220,ldots };


Am(1,5)=1,5,15,35,70,126,210,330,495,715,…{displaystyle A_{m}(1,5)=1,5,15,35,70,126,210,330,495,715,ldots }{displaystyle A_{m}(1,5)=1,5,15,35,70,126,210,330,495,715,ldots };


Am(1,6)=1,6,21,56,126,252,462,792,1287,2002,…{displaystyle A_{m}(1,6)=1,6,21,56,126,252,462,792,1287,2002,ldots }{displaystyle A_{m}(1,6)=1,6,21,56,126,252,462,792,1287,2002,ldots }.



Examples[edit]


For subindex m≥0{displaystyle mgeq 0}{displaystyle mgeq 0} the numbers are:


Examples with p=2{displaystyle p=2}p=2:




Am(2,1)=1,1,2,5,14,42,132,429,1430,4862,…{displaystyle A_{m}(2,1)=1,1,2,5,14,42,132,429,1430,4862,ldots }{displaystyle A_{m}(2,1)=1,1,2,5,14,42,132,429,1430,4862,ldots } OEIS: A000108, known as the Catalan Numbers

Am(2,2)=1,2,5,14,42,132,429,1430,4862,16796,…=Am+1(2,1){displaystyle A_{m}(2,2)=1,2,5,14,42,132,429,1430,4862,16796,ldots =A_{m+1}(2,1)}{displaystyle A_{m}(2,2)=1,2,5,14,42,132,429,1430,4862,16796,ldots =A_{m+1}(2,1)}


Am(2,3)=1,3,9,28,90,297,1001,3432,11934,41990,…{displaystyle A_{m}(2,3)=1,3,9,28,90,297,1001,3432,11934,41990,ldots }{displaystyle A_{m}(2,3)=1,3,9,28,90,297,1001,3432,11934,41990,ldots } OEIS: A000245


Am(2,4)=1,4,14,48,165,572,2002,7072,25194,90440,…{displaystyle A_{m}(2,4)=1,4,14,48,165,572,2002,7072,25194,90440,ldots }{displaystyle A_{m}(2,4)=1,4,14,48,165,572,2002,7072,25194,90440,ldots } OEIS: A002057


Examples with p=3{displaystyle p=3}p=3:




Am(3,1)=1,1,3,12,55,273,1428,7752,43263,246675,…{displaystyle A_{m}(3,1)=1,1,3,12,55,273,1428,7752,43263,246675,ldots }{displaystyle A_{m}(3,1)=1,1,3,12,55,273,1428,7752,43263,246675,ldots } OEIS: A001764


Am(3,2)=1,2,7,30,143,728,3876,21318,120175,690690,…{displaystyle A_{m}(3,2)=1,2,7,30,143,728,3876,21318,120175,690690,ldots }{displaystyle A_{m}(3,2)=1,2,7,30,143,728,3876,21318,120175,690690,ldots } OEIS: A006013

Am(3,3)=1,3,12,55,273,1428,7752,43263,246675,1430715,…=Am+1(3,1){displaystyle A_{m}(3,3)=1,3,12,55,273,1428,7752,43263,246675,1430715,ldots =A_{m+1}(3,1)}{displaystyle A_{m}(3,3)=1,3,12,55,273,1428,7752,43263,246675,1430715,ldots =A_{m+1}(3,1)}


Am(3,4)=1,4,18,88,455,2448,13566,76912,444015,2601300,…{displaystyle A_{m}(3,4)=1,4,18,88,455,2448,13566,76912,444015,2601300,ldots }{displaystyle A_{m}(3,4)=1,4,18,88,455,2448,13566,76912,444015,2601300,ldots } OEIS: A006629


Examples with p=4{displaystyle p=4}p=4:




Am(4,1)=1,1,4,22,140,969,7084,53820,420732,3362260,…{displaystyle A_{m}(4,1)=1,1,4,22,140,969,7084,53820,420732,3362260,ldots }{displaystyle A_{m}(4,1)=1,1,4,22,140,969,7084,53820,420732,3362260,ldots } OEIS: A002293


Am(4,2)=1,2,9,52,340,2394,17710,135720,1068012,8579560,…{displaystyle A_{m}(4,2)=1,2,9,52,340,2394,17710,135720,1068012,8579560,ldots }{displaystyle A_{m}(4,2)=1,2,9,52,340,2394,17710,135720,1068012,8579560,ldots } OEIS: A069271


Am(4,3)=1,3,15,91,612,4389,32890,254475,2017356,16301164,…{displaystyle A_{m}(4,3)=1,3,15,91,612,4389,32890,254475,2017356,16301164,ldots }{displaystyle A_{m}(4,3)=1,3,15,91,612,4389,32890,254475,2017356,16301164,ldots } OEIS: A006632

Am(4,4)=1,4,22,140,969,7084,53820,420732,3362260,27343888,…=Am+1(4,1){displaystyle A_{m}(4,4)=1,4,22,140,969,7084,53820,420732,3362260,27343888,ldots =A_{m+1}(4,1)}{displaystyle A_{m}(4,4)=1,4,22,140,969,7084,53820,420732,3362260,27343888,ldots =A_{m+1}(4,1)}



Algebra[edit]



Recurrence[edit]



Am(p,r)=Am(p,r−1)+Am−1(p,p+r−1){displaystyle A_{m}(p,r)=A_{m}(p,r-1)+A_{m-1}(p,p+r-1)}{displaystyle A_{m}(p,r)=A_{m}(p,r-1)+A_{m-1}(p,p+r-1)} equation (1)

This means in particular that from



Am(p,0)=0{displaystyle A_{m}(p,0)=0}{displaystyle A_{m}(p,0)=0} equation (2)

and



A0(p,r)=1{displaystyle A_{0}(p,r)=1}{displaystyle A_{0}(p,r)=1} equation (3)

one can generate all other Fuss–Catalan numbers if p is an integer.


Riordan (see references) obtains a convolution type of recurrence:



Am(p,s+r)=∑k=0mAk(p,r)Am−k(p,s){displaystyle A_{m}(p,s+r)=sum _{k=0}^{m}A_{k}(p,r)A_{m-k}(p,s)}{displaystyle A_{m}(p,s+r)=sum _{k=0}^{m}A_{k}(p,r)A_{m-k}(p,s)} equation(4)


Generating Function[edit]


Paraphrasing the Densities of the Raney distributions [2] paper, let the ordinary generating function with respect to the index m be defined as follows:



Bp,r(z):=∑m=0∞Am(p,r)zm{displaystyle B_{p,r}(z):=sum _{m=0}^{infty }A_{m}(p,r)z^{m}}{displaystyle B_{p,r}(z):=sum _{m=0}^{infty }A_{m}(p,r)z^{m}} equation (5).

Looking at equations (1) and (2), when r=1 it follows that



Am(p,p)=Am+1(p,1){displaystyle A_{m}(p,p)=A_{m+1}(p,1)}{displaystyle A_{m}(p,p)=A_{m+1}(p,1)} equation (6).

Also note this result can be derived by similar substitutions into the other formulas representation, such as the Gamma ratio representation at the top of this article. Using (6) and substituting into (5) an equivalent representation expressed as a generating function can be formulated as



Bp,1(z)=1+zBp,p(z){displaystyle B_{p,1}(z)=1+zB_{p,p}(z)}{displaystyle B_{p,1}(z)=1+zB_{p,p}(z)}.

Finally, extending this result by using Lambert's equivalence



Bp,1(z)r=Bp,r(z){displaystyle B_{p,1}(z)^{r}=B_{p,r}(z)}{displaystyle B_{p,1}(z)^{r}=B_{p,r}(z)}.

The following result can be derived for the ordinary generating function for all the Fuss-Catalan sequences.



Bp,r(z)=[1+zBp,r(z)p/r]r{displaystyle B_{p,r}(z)=[1+zB_{p,r}(z)^{p/r}]^{r}}{displaystyle B_{p,r}(z)=[1+zB_{p,r}(z)^{p/r}]^{r}}.


Recursion Representation[edit]


Recursion forms of this are as follows:
The most obvious form is:


Am(p,r)=m−1m(mp+r−1m−1)((m−1)p+r−1m−2)Am−1(p,r){displaystyle A_{m}(p,r)={frac {m-1}{m}}{frac {binom {mp+r-1}{m-1}}{binom {(m-1)p+r-1}{m-2}}}A_{m-1}(p,r)}{displaystyle A_{m}(p,r)={frac {m-1}{m}}{frac {binom {mp+r-1}{m-1}}{binom {(m-1)p+r-1}{m-2}}}A_{m-1}(p,r)}

Also, a less obvious form is


Am(p,r)=(m−1)p+rm(mp+r−1p−1)(m(p−1)+rp−1)Am−1(p,r){displaystyle A_{m}(p,r)={frac {(m-1)p+r}{m}}{frac {binom {mp+r-1}{p-1}}{binom {m(p-1)+r}{p-1}}}A_{m-1}(p,r)}{displaystyle A_{m}(p,r)={frac {(m-1)p+r}{m}}{frac {binom {mp+r-1}{p-1}}{binom {m(p-1)+r}{p-1}}}A_{m-1}(p,r)}


Alternate Representations[edit]


In some problems it is easier to use different formula configurations or variations. Below are a two examples using just the binomial function:


Am(p,r)≡rmp+r(mp+rm)=rm(p−1)+r(mp+r−1m)=rm(mp+r−1m−1){displaystyle A_{m}(p,r)equiv {frac {r}{mp+r}}{binom {mp+r}{m}}={frac {r}{m(p-1)+r}}{binom {mp+r-1}{m}}={frac {r}{m}}{binom {mp+r-1}{m-1}}}{displaystyle A_{m}(p,r)equiv {frac {r}{mp+r}}{binom {mp+r}{m}}={frac {r}{m(p-1)+r}}{binom {mp+r-1}{m}}={frac {r}{m}}{binom {mp+r-1}{m-1}}}

These variants can be converted into product, Gamma or Factorial representations too.



See also[edit]



  • Binomial coefficient

  • Binomial Distribution

  • Catalan number

  • Dyck language

  • Pascal's triangle



References[edit]





  1. ^ Clark, David (1996). "Compact Tries". Compact Pat Trees (Thesis). p. 34..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ Mlotkowski, Wojciech; Penson, Karol A.; Zyczkowski, Karol (2013). "Densities of the Raney distributions". Docum. Mathm. 18: 1593–1596. arXiv:1211.7259. Bibcode:2012arXiv1211.7259M.





  • Fuss, N. I. (1791). "Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat". Nova Acta Academiae Scientiarum Imperialis Petropolitanae. 9: 243–251.


  • Riordan, J. (1968). Combinatorial identities. Wiley. ISBN 978-0471722755.


  • Bisch, Dietmar; Jones, Vaughan (1997). "Algebras associated to intermediate subfactors". Invent. Math. 128 (1): 89–157. Bibcode:1997InMat.128...89J. doi:10.1007/s002220050137.


  • Przytycki, Jozef H.; Sikora, Adam S. (2000). "Polygon dissections and Euler, Fuss, Kirkman , and Cayley Numbers". J. Combinat. Theory A. 92: 68–76. doi:10.1006/jcta.1999.3042.


  • Aval, Jean-Christophe (2008). "Multivariate Fuss-Catalan numbers". Discrete Math. 20 (308): 4660–4669. doi:10.1016/j.disc.2007.08.100.


  • Alexeev, N.; Götze, F; Tikhomirov, A. (2010). "Asymptotic distribution of singular values of powers of random matrices". Lith. Math. J. 50 (2): 121–132. arXiv:1012.2743. doi:10.1007/s10986-010-9074-4.


  • Mlotkowski, Wojciech (2010). "Fuss-Catalan Numbers in noncommutative probability". Docum. Mathm. 15: 939–955.


  • Penson, Karol A.; Zyczkowski, Karol (2011). "Product of Ginibre matrices: Fuss-Catalan and Raney distributions". Phys. Rev. E. 83 (6): 061118. arXiv:1103.3453. Bibcode:2011PhRvE..83f1118P. doi:10.1103/PhysRevE.83.061118. PMID 21797313.


  • Gordon, Ian G.; Griffeth, Stephen (2012). "Catalan numbers for complex reflection groups". Am. J. Math. 134 (6): 1491–1502. arXiv:0912.1578. doi:10.1353/ajm.2012.0047.


  • Mlotkowski, W.; Penson, K. A. (2015). "A Fuss-type family of positive definite sequences". arXiv:1507.07312 [math.PR].


  • He, Tian-Xiao; Shapiro, Louis W. (2017). "Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group". Lin. Alg. Applic. 532: 25–42. doi:10.1016/j.laa.2017.06.025.











Retrieved from "https://en.wikipedia.org/w/index.php?title=Fuss–Catalan_number&oldid=866625483"





Navigation menu

























(window.RLQ=window.RLQ||).push(function(){mw.config.set({"wgPageParseReport":{"limitreport":{"cputime":"0.448","walltime":"0.972","ppvisitednodes":{"value":1331,"limit":1000000},"ppgeneratednodes":{"value":0,"limit":1500000},"postexpandincludesize":{"value":105827,"limit":2097152},"templateargumentsize":{"value":1014,"limit":2097152},"expansiondepth":{"value":7,"limit":40},"expensivefunctioncount":{"value":5,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":24235,"limit":5000000},"entityaccesscount":{"value":3,"limit":400},"timingprofile":["100.00% 527.114 1 -total"," 41.44% 218.455 1 Template:Underlinked"," 38.36% 202.186 1 Template:Ambox"," 25.23% 133.010 1 Template:Reflist"," 20.44% 107.733 10 Template:Cite_journal"," 16.29% 85.853 1 Template:Cite_thesis"," 15.51% 81.731 7 Template:Navbox"," 10.98% 57.874 1 Template:Classes_of_natural_numbers"," 2.55% 13.433 1 Template:Cite_arxiv"," 2.19% 11.532 1 Template:Icon"]},"scribunto":{"limitreport-timeusage":{"value":"0.218","limit":"10.000"},"limitreport-memusage":{"value":4783305,"limit":52428800}},"cachereport":{"origin":"mw1238","timestamp":"20181129151126","ttl":1900800,"transientcontent":false}}});mw.config.set({"wgBackendResponseTime":102,"wgHostname":"mw1330"});});
-

Popular posts from this blog

Florida Star v. B. J. F.

Image
Florida Star v. B. J. F. From Wikipedia, the free encyclopedia Jump to navigation Jump to search United States Supreme Court case Florida Star v. B. J. F. Supreme Court of the United States Argued March 21, 1989 Decided June 21, 1989 Full case name The Florida Star v. B. J. F. Citations 491 U.S. 524 ( more ) 109 S. Ct. 2603; 105 L. Ed. 2d 443; 1989 U.S. LEXIS 3120; 57 U.S.L.W. 4816; 16 Media L. Rep. 1801 Prior history The Florida Star v. B.J.F., 530 So.2d 286 (1988) Supreme Court of Florida; Florida Star v. B.J.F., 499 So.2d 883 (1986) Fla. Dist. Court of Appeals Holding Florida Stat. § 794.03 is unconstitutional to the extent it makes the truthful reporting of information that was a matter of public record unlawful, as it violates the First Amendment. Court membership Chief Justice William Rehnquist Associate Justices William J. Brennan Jr.  · Byron White Thurgood Marshall  · Harry Blac...
Read more

Danny Elfman

Image
Danny Elfman From Wikipedia, the free encyclopedia Jump to navigation Jump to search Danny Elfman Elfman at the 2010 San Diego Comic-Con Born Daniel Robert Elfman ( 1953-05-29 ) May 29, 1953 (age 65) Los Angeles, California, U.S. Spouse(s) Bridget Fonda ( m.  2003) Children 1 Musical career Genres Rock [1] ska [2] new wave film music video game music Occupation(s) Composer, singer, songwriter, record producer Instruments Trombone guitar percussion vocals keyboards [3] Years active 1972–present Associated acts Oingo Boingo James Newton Howard Daniel Robert Elfman (born May 29, 1953) is an American composer, singer, songwriter, and record producer. Elfman first became known for being the lead singer and songwriter for the band Oingo Boingo from 1974 to 1995. He is well known for scoring films and television shows, particularly his frequent collabora...
Read more

Lugert, Oklahoma

Image
Unincorporated community in Oklahoma, United States Lugert, Oklahoma Unincorporated community Foundations from the town of Lugert at the bottom of Lake Altus-Lugert Lugert, Oklahoma Location within the state of Oklahoma Coordinates: 34°53′45″N 99°16′31″W  /  34.89583°N 99.27528°W  / 34.89583; -99.27528 Coordinates: 34°53′45″N 99°16′31″W  /  34.89583°N 99.27528°W  / 34.89583; -99.27528 Country United States State Oklahoma County Kiowa Time zone UTC-6 (Central (CST))  • Summer (DST) UTC-5 (CDT) GNIS feature ID 1100597 Lugert is an unincorporated community in Kiowa County, Oklahoma. [1] The town of Lugert was founded in 1901 on 80 acres (320,000 m 2 ). In the town, there was a general store that housed the Post office and sold dry goods, school supplies, groceries, harnesses, axes and much more. It was named for Frank Lugert, who had moved to the area in 1898. Lugert owned land where the town was sited and also owned the general st...
Read more