The charm bracelet problem and its applications

Stockmeyer, Paul K.

doi:10.1007/BFb0066456

Paul K. Stockmeyer¹

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 406))

2735 Accesses
1 Citations
3 Altmetric

Abstract

The necklace problem has proved to be both a sound pedagogical device in teaching enumeration theory and a valuable counting tool with several graphical applications. In this paper we solve the more general charm bracelet problem and provide two applications for which the necklace problem in not sufficient.

We set the stage in Section 1 by providing a brief review of the necklace problem. This serves as a basis for comparison in Section 2, where we discuss the charm bracelet problem and derive its solution. Sections 3 and 4 contain nontrivial graphical applications of the results of Section 2.

Definitions for all graphical terms and concepts can be found in [3]. For further background and broader treatment of topics of an enumerative nature, [5] should be consulted.

This research was supported by the Office of Naval Research under contract N00014-73-A-0374-0001, NR044-459. Reproduction in whole or in part is permitted for any purpose of the United States Government.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic

$34.99 /Month

Get 10 units per month
Download Article/Chapter or eBook
1 Unit = 1 Article or 1 Chapter
Cancel anytime

Buy Now

Chapter: USD 29.95; Price excludes VAT (USA)

eBook: USD 39.99; Price excludes VAT (USA)

Softcover Book: USD 54.99; Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

') var buybox = document.querySelector("[data-id=id_"+ timestamp +"]").parentNode var buyingOptions = buybox.querySelectorAll(".buying-option") ;[].slice.call(buyingOptions).forEach(initCollapsibles) // springerPlus roll out 10% starts here var springerPlusGroup = setLocalStorageSpringerPlus(); var rollOutSpringerPlus = springerPlusGroup === "B" function setLocalStorageSpringerPlus() { var selectUserKey = "springerPlusRollOut"; var springerPlusGroup = "X"; if (!window.localStorage) return springerPlusGroup; try { var selectUserValue = window.localStorage.getItem(selectUserKey) springerPlusGroup = selectUserValue || randomDistributionSpringerPlus(selectUserKey) } catch (err) { console.log(err) } return springerPlusGroup; } function randomDistributionSpringerPlus(selectUserKey) { var randomGroup = Math.random() < 0.7 ? "A" : "B" window.localStorage.setItem(selectUserKey, randomGroup) return randomGroup } if (rollOutSpringerPlus) { revealSpringerPlus(); } function revealSpringerPlus() { if(buybox) { document.querySelectorAll(".c-springer-plus").forEach(function(node) { node.style.display = "block" }) } } //springerPlus ends here var buyboxMaxSingleColumnWidth = 480 function initCollapsibles(subscription, index) { var toggle = subscription.querySelector(".buying-option-price") subscription.classList.remove("expanded") var form = subscription.querySelector(".buying-option-form") var priceInfo = subscription.querySelector(".price-info") var buyingOption = toggle.parentElement if (toggle && form && priceInfo) { toggle.setAttribute("role", "button") toggle.setAttribute("tabindex", "0") toggle.addEventListener("click", function (event) { var expandedBuyingOptions = buybox.querySelectorAll(".buying-option.expanded") var buyboxWidth = buybox.offsetWidth ;[].slice.call(expandedBuyingOptions).forEach(function(option) { if (buyboxWidth <= buyboxMaxSingleColumnWidth && option != buyingOption) { hideBuyingOption(option) } }) var expanded = toggle.getAttribute("aria-expanded") === "true" || false toggle.setAttribute("aria-expanded", !expanded) form.hidden = expanded if (!expanded) { buyingOption.classList.add("expanded") } else { buyingOption.classList.remove("expanded") } priceInfo.hidden = expanded }, false) } } function hideBuyingOption(buyingOption) { var toggle = buyingOption.querySelector(".buying-option-price") var form = buyingOption.querySelector(".buying-option-form") var priceInfo = buyingOption.querySelector(".price-info") toggle.setAttribute("aria-expanded", false) form.hidden = true buyingOption.classList.remove("expanded") priceInfo.hidden = true } function initKeyControls() { document.addEventListener("keydown", function (event) { if (document.activeElement.classList.contains("buying-option-price") && (event.code === "Space" || event.code === "Enter")) { if (document.activeElement) { event.preventDefault() document.activeElement.click() } } }, false) } function initialStateOpen() { var buyboxWidth = buybox.offsetWidth ;[].slice.call(buybox.querySelectorAll(".buying-option")).forEach(function (option, index) { var toggle = option.querySelector(".buying-option-price") var form = option.querySelector(".buying-option-form") var priceInfo = option.querySelector(".price-info") if (buyboxWidth > buyboxMaxSingleColumnWidth) { toggle.click() } else { if (index === 0) { toggle.click() } else { toggle.setAttribute("aria-expanded", "false") form.hidden = "hidden" priceInfo.hidden = "hidden" } } }) } initialStateOpen() if (window.buyboxInitialised) return window.buyboxInitialised = true initKeyControls() })()

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

References

Burnside, W., Theory of Groups of Finite Order. Second Edition, Cambridge University Press, 1911. Reprinted Dover, 1955, New York.
Google Scholar
Guy, R. K., "Dissecting a Polygon into Triangles", Research Report, University of Calgary, 1960.
Google Scholar
Harary, F., Graph Theory. Addison-Wesley, 1969, Reading.
Google Scholar
Harary, F., "Enumeration Under Group Action: Unsolved Graphical Enumeration Problems, IV." J. Comb. Theory, 8 (1970) 1–11.
Article MATH Google Scholar
Harary, F., and Palmer, E. M., Graphical Enumeration, Academic Press, 1973, New York.
MATH Google Scholar
Harary, F., and Prins, G., "The Number of Homomorphically Irreducible Trees and Other Species", Acta Math. 101 (1959) 141–162.
Article MathSciNet MATH Google Scholar
Harary, F., Prins, G., Tutte, W. t., "The Number of Plane Trees", Indag. Math 26 (1964) 319–329.
MathSciNet Google Scholar
Harary, F., and Robinson, R. W., "The Number of Achiral Trees", J. Reine Angew. Math., to appear.
Google Scholar
Moon, J. W., and Moser, L., "Triangular Dissections of n-gons" Canad. Math. Bull. 6 (1963) 175–177.
Article MathSciNet MATH Google Scholar
Otter, R., "The Number of Trees", Ann. of Math. 49 (1948) 583–599.
Article MathSciNet MATH Google Scholar
Pólya, G., "Kombinatorische Anzehlbestimmungen für Gruppen, Graphen und chemische Verbindungen", Acta Math. 68 (1937) 145–254.
Article Google Scholar

Download references

Author information

Authors and Affiliations

College of William and Mary, USA
Paul K. Stockmeyer

Authors

Paul K. Stockmeyer
View author publications
You can also search for this author in PubMed Google Scholar

Editor information

Ruth A. Bari Frank Harary

Rights and permissions

Reprints and permissions

Copyright information

About this paper

Cite this paper

Stockmeyer, P.K. (1974). The charm bracelet problem and its applications. In: Bari, R.A., Harary, F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066456

Download citation

DOI: https://doi.org/10.1007/BFb0066456
Published: 26 August 2006
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06854-9
Online ISBN: 978-3-540-37809-9
eBook Packages: Springer Book Archive

Publish with us

Policies and ethics