document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210232>Question 1210232</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Recall that a partition of a positive integer n means a way of writing n as the sum of some positive integers, where the order of the parts does not matter.  For example, there are five partitions of 4:\r<BR>\n" );
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document.write( "4\r<BR>\n" );
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document.write( "3 + 1\r<BR>\n" );
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document.write( "2 + 2\r<BR>\n" );
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document.write( "2 + 1 + 1\r<BR>\n" );
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document.write( "1 + 1 + 1 + 1\r<BR>\n" );
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document.write( "How many partitions of 17 are there that have at least three parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to 4, 3, 2, and 1?\r<BR>\n" );
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document.write( "The partition 17 = 7 + 4 + 3 + 2 + 1 is one such partition.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851704 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851704\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Solution:<BR>\n" );
document.write( "Let a partition of 17 with at least three parts be $\lambda_1 \ge \lambda_2 \ge \lambda_3 \ge \dots \ge \lambda_k > 0$, where $k \ge 3$ and $\sum_{i=1}^k \lambda_i = 17$.<BR>\n" );
document.write( "The conditions on the largest, second-largest, third-largest, and fourth-largest parts are:<BR>\n" );
document.write( "$\lambda_1 \ge 4$<BR>\n" );
document.write( "$\lambda_2 \ge 3$<BR>\n" );
document.write( "$\lambda_3 \ge 2$<BR>\n" );
document.write( "$\lambda_4 \ge 1$ (if $k \ge 4$)\r<BR>\n" );
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document.write( "Consider a partition of 17 into $k$ parts satisfying these conditions.<BR>\n" );
document.write( "Let $\lambda_1 = a_1 + 4, \lambda_2 = a_2 + 3, \lambda_3 = a_3 + 2, \lambda_i = a_i + 1$ for $i \ge 4$, where $a_1 \ge a_2 \ge a_3 \ge a_4 \ge \dots \ge a_k \ge 0$.\r<BR>\n" );
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document.write( "Case 1: Exactly 3 parts ($k=3$)<BR>\n" );
document.write( "$\lambda_1 \ge \lambda_2 \ge \lambda_3 > 0$<BR>\n" );
document.write( "$\lambda_1 \ge 4, \lambda_2 \ge 3, \lambda_3 \ge 2$<BR>\n" );
document.write( "Let $\lambda_1 = x+4, \lambda_2 = y+3, \lambda_3 = z+2$, where $x \ge y \ge z \ge 0$.<BR>\n" );
document.write( "$(x+4) + (y+3) + (z+2) = 17 \implies x+y+z = 8$.<BR>\n" );
document.write( "Partitions of 8 into 3 non-negative integers:<BR>\n" );
document.write( "(8, 0, 0), (7, 1, 0), (6, 2, 0), (6, 1, 1), (5, 3, 0), (5, 2, 1), (4, 4, 0), (4, 3, 1), (4, 2, 2), (3, 3, 2).<BR>\n" );
document.write( "These correspond to partitions of 17: 12+3+2, 11+4+2, 10+5+2, 10+4+3, 9+6+2, 9+5+3, 8+7+2, 8+6+3, 8+5+4, 7+6+4. (10 partitions)\r<BR>\n" );
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document.write( "Case 2: Exactly 4 parts ($k=4$)<BR>\n" );
document.write( "$\lambda_1 \ge 4, \lambda_2 \ge 3, \lambda_3 \ge 2, \lambda_4 \ge 1$.<BR>\n" );
document.write( "Let $\lambda_1 = x+4, \lambda_2 = y+3, \lambda_3 = z+2, \lambda_4 = w+1$, where $x \ge y \ge z \ge w \ge 0$.<BR>\n" );
document.write( "$(x+4) + (y+3) + (z+2) + (w+1) = 17 \implies x+y+z+w = 7$.<BR>\n" );
document.write( "Partitions of 7 into 4 non-negative integers:<BR>\n" );
document.write( "(7, 0, 0, 0), (6, 1, 0, 0), (5, 2, 0, 0), (5, 1, 1, 0), (4, 3, 0, 0), (4, 2, 1, 0), (4, 1, 1, 1), (3, 3, 1, 0), (3, 2, 2, 0), (3, 2, 1, 1), (2, 2, 2, 1). (11 partitions)\r<BR>\n" );
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document.write( "Case 3: Exactly 5 parts ($k=5$)<BR>\n" );
document.write( "$\lambda_1 \ge 4, \lambda_2 \ge 3, \lambda_3 \ge 2, \lambda_4 \ge 1, \lambda_5 \ge 1$.<BR>\n" );
document.write( "$x+4+y+3+z+2+w+1+v+1 = 17 \implies x+y+z+w+v = 6$. (7 partitions)\r<BR>\n" );
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document.write( "Case 4: Exactly 6 parts ($k=6$)<BR>\n" );
document.write( "Sum of min parts = $4+3+2+1+1+1 = 12$. $x+y+z+w+v+u = 5$. (5 partitions)\r<BR>\n" );
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document.write( "Case 5: Exactly 7 parts ($k=7$)<BR>\n" );
document.write( "Sum of min parts = $4+3+2+1+1+1+1 = 13$. $x+y+z+w+v+u+t = 4$. (5 partitions)\r<BR>\n" );
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document.write( "Case 6: Exactly 8 parts ($k=8$)<BR>\n" );
document.write( "Sum of min parts = $4+3+2+1+1+1+1+1 = 14$. $x+y+z+w+v+u+t+s = 3$. (3 partitions)\r<BR>\n" );
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document.write( "Case 7: Exactly 9 parts ($k=9$)<BR>\n" );
document.write( "Sum of min parts = $4+3+2+1+1+1+1+1+1 = 15$. $x+...+r = 2$. (2 partitions)\r<BR>\n" );
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document.write( "Case 8: Exactly 10 parts ($k=10$)<BR>\n" );
document.write( "Sum of min parts = $4+3+2+1+1+1+1+1+1+1 = 16$. $x+...+q = 1$. (1 partition)\r<BR>\n" );
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document.write( "Case 9: Exactly 11 parts ($k=11$)<BR>\n" );
document.write( "Sum of min parts = $4+3+2+1+1+1+1+1+1+1+1 = 17$. $x+... = 0$. (1 partition)\r<BR>\n" );
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document.write( "Total = $10 + 11 + 7 + 5 + 5 + 3 + 2 + 1 + 1 = 45$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{45}$ </SPAN>\n" );
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