{"id":51,"date":"2024-02-20T10:21:23","date_gmt":"2024-02-20T10:21:23","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=51"},"modified":"2024-02-22T20:30:26","modified_gmt":"2024-02-22T20:30:26","slug":"catalan-triangle-and-clebsch-gordan-multiplicities","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/02\/20\/catalan-triangle-and-clebsch-gordan-multiplicities\/","title":{"rendered":"Catalan Triangle and Clebsch-Gordan Multiplicities"},"content":{"rendered":"\n

Most magnetic resonance spectroscopists will invariably have seen Pascal’s triangle<\/a> (i.e. the triangle of binomial coefficients) in introductory undergraduate courses or school curricula:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Pascal’s triangle is widely used as a pedagogical tool to explain first-order multiplet patterns<\/a>.<\/p>\n\n\n\n

However, magnetic resonance is full of other, less well-known combinatorial structures. One of the most useful is closely related: a Catalan triangle<\/a>* (so named due to the leftmost columns giving the Catalan numbers<\/em><\/a>):<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

In basic terms, this Catalan triangle (which adds up just like Pascal’s triangle) provides the distribution of eigenstates in a symmetric N<\/em>-spin-1\/2 system, immensely <\/em>simplifying the treatment of multispin problems. For example, an A2<\/sub> spin system such as H2<\/sub><\/a> can be treated as a sum of 1 spin-0 (singlet) and 1 spin-1 (triplet) particle. An A4<\/sub> spin system such as CH4<\/sub><\/a> can be treated as a sum of 2 spin-0, 3 spin-1, and 1 spin-2 particles:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Deep in the annals of NMR theory, the coefficients of this Catalan triangle are also known as the “Clebsch-Gordan multiplicities<\/a>“. <\/p>\n\n\n\n

The old NMR texts referred to the above decomposition as the composite particle method<\/a> showing that this was a much <\/em>easier way to treat multispin systems than going Rambo<\/a> and invoking symmetry groups, character tables, molecular rotation, and what have you. The simple triangle itself appears in some form in early references such as Grimley’s paper<\/a> (1963) and Corio’s famous book<\/a> (1967):<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

It is rather unfortunate that in today’s literature, there is an apparent insistence on obfuscating such basic combinatorial results by invoking unnecessarily complicated, apocryphal chains of group theoretical arguments – behavior that early quantum physicists would have called gruppenpest<\/a><\/em>.<\/p>\n\n\n\n

A beautiful recent example of the appearance of the coefficients of this Catalan triangle in an unexpected context is this excellent paper<\/a>, which rigorously explores weighted multiplets; for example, the intensity ratios of the simple z<\/em>1<\/sup> multiplets (generated by the INEPT sequence) are given by a “mirrored” Catalan triangle:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Historically, this Catalan triangle can be traced back to Wigner’s timeless book (1959)<\/a> (where it appears as an expression) and the classic combinatorics paper<\/a> of Forder (1961), although there are surely older, less relevant appearances in the literature.<\/p>\n\n\n\n

*Note: unfortunately, due to the ubiquity of Catalan numbers<\/em><\/a> in mathematics, “Catalan triangle” and “Catalan’s triangle” have been used in the literature to refer to several <\/em>different number triangles, generating immense confusion. Since there are many Catalan triangles, care should be taken with terms such as “Catalan’s triangle” or “the <\/em>Catalan triangle” without an appropriate reference, and one should use terms such as “a <\/em>Catalan triangle” or simply “triangle of Clebsch-Gordan multiplicites”. I personally like “Catalan-type triangle”.<\/p>\n\n\n\n

\u2020<\/strong>There are some pretty results, such as the number of spin-0 (singlet) states in a (2N<\/em>)-spin-1\/2 system being given by the N<\/em>-th Catalan number. This is also true for the number of spin-1\/2 states in a (2N-1)<\/em>-spin-1\/2 system.<\/p>\n","protected":false},"excerpt":{"rendered":"

Most magnetic resonance spectroscopists will invariably have seen Pascal’s triangle (i.e. the triangle of binomial coefficients) in introductory undergraduate courses or school curricula: Pascal’s triangle is widely used as a pedagogical tool to explain first-order multiplet patterns. However, magnetic resonance is full of other, less well-known combinatorial structures. One of the most useful is closely […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16,7],"tags":[],"class_list":["post-51","post","type-post","status-publish","format-standard","hentry","category-combinatorial-aspects-of-magnetic-resonance","category-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/51","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/comments?post=51"}],"version-history":[{"count":1,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/51\/revisions"}],"predecessor-version":[{"id":59,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/51\/revisions\/59"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=51"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=51"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}