History of Monomial symmetric polynomials $m_\lambda$

 - combinatorics - algebra+- bases for symmetric polynomials Programs:   program-sage:

       \ [3, 2])    # a partition of 5, past this table" Numbers:-- params:-    n: '1'-    lambda: '1'-  number: x1-- params:-    n: '1'-    lambda: '2'-  number: x1^2-- params:-    n: '1'-    lambda: '3'-  number: x1^3-- params:-    n: '1'-    lambda: '4'-  number: x1^4-- params:-    n: '2'-    lambda: '1'-  number: x1 + x2-- params:-    n: '2'-    lambda: '2'-  number: x1^2 + x2^2-- params:-    n: '2'-    lambda: 1,1-  number: x1*x2-- params:-    n: '2'-    lambda: '3'-  number: x1^3 + x2^3-- params:-    n: '2'-    lambda: 2,1-  number: x1^2*x2 + x1*x2^2-- params:-    n: '2'-    lambda: '4'-  number: x1^4 + x2^4-- params:-    n: '2'-    lambda: 3,1-  number: x1^3*x2 + x1*x2^3-- params:-    n: '2'-    lambda: 2,2-  number: x1^2*x2^2-- params:-    n: '3'-    lambda: '1'-  number: x1 + x2 + x3-- params:-    n: '3'-    lambda: '2'-  number: x1^2 + x2^2 + x3^2-- params:-    n: '3'-    lambda: 1,1-  number: x1*x2 + x1*x3 + x2*x3-- params:-    n: '3'-    lambda: '3'-  number: x1^3 + x2^3 + x3^3-- params:-    n: '3'-    lambda: 2,1-  number: x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2-- params:-    n: '3'-    lambda: 1,1,1-  number: x1*x2*x3-- params:-    n: '3'-    lambda: '4'-  number: x1^4 + x2^4 + x3^4-- params:-    n: '3'-    lambda: 3,1-  number: x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3-- params:-    n: '3'-    lambda: 2,2-  number: x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2-- params:-    n: '3'-    lambda: 2,1,1-  number: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2-- params:-    n: '4'-    lambda: '1'-  number: x1 + x2 + x3 + x4-- params:-    n: '4'-    lambda: '2'-  number: x1^2 + x2^2 + x3^2 + x4^2-- params:-    n: '4'-    lambda: 1,1-  number: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4-- params:-    n: '4'-    lambda: '3'-  number: x1^3 + x2^3 + x3^3 + x4^3-- params:-    n: '4'-    lambda: 2,1-  number: x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4-    + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2-- params:-    n: '4'-    lambda: 1,1,1-  number: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4-- params:-    n: '4'-    lambda: '4'-  number: x1^4 + x2^4 + x3^4 + x4^4-- params:-    n: '4'-    lambda: 3,1-  number: x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4-    + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3-- params:-    n: '4'-    lambda: 2,2-  number: x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2-- params:-    n: '4'-    lambda: 2,1,1-  number: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4-    + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2-- params:-    n: '4'-    lambda: 1,1,1,1-  number: x1*x2*x3*x4-- params:-    n: '5'-    lambda: '1'-  number: x1 + x2 + x3 + x4 + x5-- params:-    n: '5'-    lambda: '2'-  number: x1^2 + x2^2 + x3^2 + x4^2 + x5^2-- params:-    n: '5'-    lambda: 1,1-  number: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 +-    x4*x5-- params:-    n: '5'-    lambda: '3'-  number: x1^3 + x2^3 + x3^3 + x4^3 + x5^3-- params:-    n: '5'-    lambda: 2,1-  number: x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4-    + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x1^2*x5 + x2^2*x5 + x3^2*x5 + x4^2*x5-    + x1*x5^2 + x2*x5^2 + x3*x5^2 + x4*x5^2-- params:-    n: '5'-    lambda: 1,1,1-  number: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5-    + x1*x4*x5 + x2*x4*x5 + x3*x4*x5-- params:-    n: '5'-    lambda: '4'-  number: x1^4 + x2^4 + x3^4 + x4^4 + x5^4-- params:-    n: '5'-    lambda: 3,1-  number: x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4-    + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x1^3*x5 + x2^3*x5 + x3^3*x5 + x4^3*x5-    + x1*x5^3 + x2*x5^3 + x3*x5^3 + x4*x5^3-- params:-    n: '5'-    lambda: 2,2-  number: x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2 +-    x1^2*x5^2 + x2^2*x5^2 + x3^2*x5^2 + x4^2*x5^2-- params:-    n: '5'-    lambda: 2,1,1-  number: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4-    + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2-    + x1^2*x2*x5 + x1*x2^2*x5 + x1^2*x3*x5 + x2^2*x3*x5 + x1*x3^2*x5 + x2*x3^2*x5-    + x1^2*x4*x5 + x2^2*x4*x5 + x3^2*x4*x5 + x1*x4^2*x5 + x2*x4^2*x5 + x3*x4^2*x5-    + x1*x2*x5^2 + x1*x3*x5^2 + x2*x3*x5^2 + x1*x4*x5^2 + x2*x4*x5^2 + x3*x4*x5^2-- params:-    n: '5'-    lambda: 1,1,1,1-  number: x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5+  '1':+    '1': x1+    '2': x1^2+    '3': x1^3+    '4': x1^4+  '2':+    '1': x1 + x2+    '2': x1^2 + x2^2+    1,1: x1*x2+    '3': x1^3 + x2^3+    2,1: x1^2*x2 + x1*x2^2+    '4': x1^4 + x2^4+    3,1: x1^3*x2 + x1*x2^3+    2,2: x1^2*x2^2+  '3':+    '1': x1 + x2 + x3+    '2': x1^2 + x2^2 + x3^2+    1,1: x1*x2 + x1*x3 + x2*x3+    '3': x1^3 + x2^3 + x3^3+    2,1: x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2+    1,1,1: x1*x2*x3+    '4': x1^4 + x2^4 + x3^4+    3,1: x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3+    2,2: x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2+    2,1,1: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2+  '4':+    '1': x1 + x2 + x3 + x4+    '2': x1^2 + x2^2 + x3^2 + x4^2+    1,1: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4+    '3': x1^3 + x2^3 + x3^3 + x4^3+    2,1: x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4+      + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2+    1,1,1: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4+    '4': x1^4 + x2^4 + x3^4 + x4^4+    3,1: x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4+      + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3+    2,2: x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2+    2,1,1: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4+      + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2+    1,1,1,1: x1*x2*x3*x4+  '5':+    '1': x1 + x2 + x3 + x4 + x5+    '2': x1^2 + x2^2 + x3^2 + x4^2 + x5^2+    1,1: x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 + x4*x5+    '3': x1^3 + x2^3 + x3^3 + x4^3 + x5^3+    2,1: x1^2*x2 + x1*x2^2 + x1^2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + x2^2*x4+      + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x1^2*x5 + x2^2*x5 + x3^2*x5 + x4^2*x5+      + x1*x5^2 + x2*x5^2 + x3*x5^2 + x4*x5^2+    1,1,1: x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5+      + x1*x4*x5 + x2*x4*x5 + x3*x4*x5+    '4': x1^4 + x2^4 + x3^4 + x4^4 + x5^4+    3,1: x1^3*x2 + x1*x2^3 + x1^3*x3 + x2^3*x3 + x1*x3^3 + x2*x3^3 + x1^3*x4 + x2^3*x4+      + x3^3*x4 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x1^3*x5 + x2^3*x5 + x3^3*x5 + x4^3*x5+      + x1*x5^3 + x2*x5^3 + x3*x5^3 + x4*x5^3+    2,2: x1^2*x2^2 + x1^2*x3^2 + x2^2*x3^2 + x1^2*x4^2 + x2^2*x4^2 + x3^2*x4^2 + x1^2*x5^2+      + x2^2*x5^2 + x3^2*x5^2 + x4^2*x5^2+    2,1,1: x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4+      + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2+      + x1^2*x2*x5 + x1*x2^2*x5 + x1^2*x3*x5 + x2^2*x3*x5 + x1*x3^2*x5 + x2*x3^2*x5+      + x1^2*x4*x5 + x2^2*x4*x5 + x3^2*x4*x5 + x1*x4^2*x5 + x2*x4^2*x5 + x3*x4^2*x5+      + x1*x2*x5^2 + x1*x3*x5^2 + x2*x3*x5^2 + x1*x4*x5^2 + x2*x4*x5^2 + x3*x4*x5^2+    1,1,1,1: x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5