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<title>Spt function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Spt function</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>spt function</b> (smallest parts function) is a function in <a href="Number_theory" title="Number theory">number theory</a> that counts the sum of the number of smallest parts in each <a href="Integer_partition" title="Integer partition">integer partition</a> of a positive integer. It is related to the <a href="Partition_function_(number_theory)" title="Partition function (number theory)">partition function</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The first few values of spt(<i>n</i>) are:

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<dl><dd>1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ... (sequence <span class="nowrap external"><a href="https://oeis.org/A092269" class="extiw external" title="oeis:A092269">A092269</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

<p>For example, there are five partitions of 4 (with smallest parts underlined):

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<p>These partitions have 1, 1, 2, 2, and 4 smallest parts, respectively. So spt(4) = 1 + 1 + 2 + 2 + 4 = 10.

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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>Like the partition function, spt(<i>n</i>) has a <a href="Generating_function" title="Generating function">generating function</a>. It is given by

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<annotation encoding="application/x-tex">{\displaystyle S(q)=\sum _{n=1}^{\infty }\mathrm {spt} (n)q^{n}={\frac {1}{(q)_{\infty }}}\sum _{n=1}^{\infty }{\frac {q^{n}\prod _{m=1}^{n-1}(1-q^{m})}{1-q^{n}}}}</annotation>

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</math></span><img src="./2a858d56261ef5a35764170ceb61713ba9171ab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.709ex; height:7.343ex;" alt="{\displaystyle S(q)=\sum _{n=1}^{\infty }\mathrm {spt} (n)q^{n}={\frac {1}{(q)_{\infty }}}\sum _{n=1}^{\infty }{\frac {q^{n}\prod _{m=1}^{n-1}(1-q^{m})}{1-q^{n}}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}</annotation>

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</math></span><img src="./d46d1738c63ab72453b26c686fae2568e570f27b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.049ex; height:6.843ex;" alt="{\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}" loading="lazy"></span>.

</p><p>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(q)}">

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<annotation encoding="application/x-tex">{\displaystyle S(q)}</annotation>

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</math></span><img src="./c5afeeb5ad0b7d09d9c2aa1795e6bcfd7f15e6ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle S(q)}" loading="lazy"></span> is related to a <a href="Mock_modular_form" title="Mock modular form">mock modular form</a>. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{2}(z)}">

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<annotation encoding="application/x-tex">{\displaystyle E_{2}(z)}</annotation>

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</math></span><img src="./ffee652f565d55da66bafd55481c78aeb714b664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.667ex; height:2.843ex;" alt="{\displaystyle E_{2}(z)}" loading="lazy"></span> denote the weight 2 quasi-modular <a href="Eisenstein_series" title="Eisenstein series">Eisenstein series</a> and let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \eta (z)}</annotation>

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</math></span><img src="./c9c80ad7f12439ef07ba49197ea1d72f9bbc31e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.067ex; height:2.843ex;" alt="{\displaystyle \eta (z)}" loading="lazy"></span> denote the <a href="Dedekind_eta_function" title="Dedekind eta function">Dedekind eta function</a>. Then for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=e^{2\pi iz}}">

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<annotation encoding="application/x-tex">{\displaystyle q=e^{2\pi iz}}</annotation>

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</math></span><img src="./cbd4a07533fb952a40f930a40b1d7437aad86b26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.584ex; height:3.009ex;" alt="{\displaystyle q=e^{2\pi iz}}" loading="lazy"></span>, the function

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {S}}(z):=q^{-1/24}S(q)-{\frac {1}{12}}{\frac {E_{2}(z)}{\eta (z)}}}</annotation>

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</math></span><img src="./90aa180ba257c106ed7b1c8ff6629865a7b0ecaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.976ex; height:6.509ex;" alt="{\displaystyle {\tilde {S}}(z):=q^{-1/24}S(q)-{\frac {1}{12}}{\frac {E_{2}(z)}{\eta (z)}}}" loading="lazy"></span></dd></dl>

<p>is a <a href="Mock_modular_form" title="Mock modular form">mock modular form</a> of weight 3/2 on the full <a href="Modular_group" title="Modular group">modular group</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL_{2}(\mathbb {Z} )}">

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<annotation encoding="application/x-tex">{\displaystyle SL_{2}(\mathbb {Z} )}</annotation>

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</math></span><img src="./d3b036312486fabaa4a90953a39d8b7fcb88d6dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.496ex; height:2.843ex;" alt="{\displaystyle SL_{2}(\mathbb {Z} )}" loading="lazy"></span> with multiplier system <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\eta }^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{\eta }^{-1}}</annotation>

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</math></span><img src="./bf867ed3b66c6beb8b1b567b551ac36c9b083228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.788ex; height:3.343ex;" alt="{\displaystyle \chi _{\eta }^{-1}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\eta }}">

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</math></span><img src="./f2618b363c093ece36d8c5f7a8723ef430d52574.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.514ex; height:2.343ex;" alt="{\displaystyle \chi _{\eta }}" loading="lazy"></span> is the multiplier system for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \eta (z)}</annotation>

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</math></span><img src="./c9c80ad7f12439ef07ba49197ea1d72f9bbc31e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.067ex; height:2.843ex;" alt="{\displaystyle \eta (z)}" loading="lazy"></span>.

</p><p>While a closed formula is not known for spt(<i>n</i>), there are Ramanujan-like <a href="Ramanujan's_congruences" title="Ramanujan's congruences">congruences</a> including

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {spt} (5n+4)\equiv 0\mod (5)}</annotation>

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</math></span><img src="./0f4eb2f6bb2d1c56c51c3d087b3fb0247a7cdae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.59ex; height:2.843ex;" alt="{\displaystyle \mathrm {spt} (5n+4)\equiv 0\mod (5)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {spt} (7n+5)\equiv 0\mod (7)}</annotation>

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</math></span><img src="./d8360101186764a459fd1563352d338347560b49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.59ex; height:2.843ex;" alt="{\displaystyle \mathrm {spt} (7n+5)\equiv 0\mod (7)}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {spt} (13n+6)\equiv 0\mod (13).}</annotation>

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</math></span><img src="./3e0f5e69c9dbcbc607ce05c9c08be9f02a79b3c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.562ex; height:2.843ex;" alt="{\displaystyle \mathrm {spt} (13n+6)\equiv 0\mod (13).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFAndrews2008" class="citation journal cs1"><a href="George_Andrews_(mathematician)" title="George Andrews (mathematician)">Andrews, George E.</a> (2008-11-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/CRELLE.2008.083/html">"The number of smallest parts in the partitions of n"</a></span>. <i>Journal für die Reine und Angewandte Mathematik</i>. <b>2008</b> (624): <span class="nowrap">133–</span>142. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2FCRELLE.2008.083">10.1515/CRELLE.2008.083</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1435-5345">1435-5345</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123142859">123142859</a>.</cite></span>

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