History of Complete homogeneous symmetric polynomials $h_k$

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

       \ 4)                # the next degree after this table" Numbers:-- params:-    n: '1'-    k: '1'-  number: x1-- params:-    n: '1'-    k: '2'-  number: x1^2-- params:-    n: '1'-    k: '3'-  number: x1^3-- params:-    n: '2'-    k: '1'-  number: x1 + x2-- params:-    n: '2'-    k: '2'-  number: x1^2 + x1*x2 + x2^2-- params:-    n: '2'-    k: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3-- params:-    n: '3'-    k: '1'-  number: x1 + x2 + x3-- params:-    n: '3'-    k: '2'-  number: x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2-- params:-    n: '3'-    k: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2-    + x2*x3^2 + x3^3-- params:-    n: '4'-    k: '1'-  number: x1 + x2 + x3 + x4-- params:-    n: '4'-    k: '2'-  number: x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2-- params:-    n: '4'-    k: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2-    + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4-    + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3-- params:-    n: '5'-    k: '1'-  number: x1 + x2 + x3 + x4 + x5-- params:-    n: '5'-    k: '2'-  number: x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2-    + x1*x5 + x2*x5 + x3*x5 + x4*x5 + x5^2-- params:-    n: '5'-    k: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2-    + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4-    + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3 + x1^2*x5 + x1*x2*x5 + x2^2*x5 + x1*x3*x5-    + x2*x3*x5 + x3^2*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x4^2*x5 + x1*x5^2 + x2*x5^2-    + x3*x5^2 + x4*x5^2 + x5^3-- params:-    n: '6'-    k: '1'-  number: x1 + x2 + x3 + x4 + x5 + x6-- params:-    n: '6'-    k: '2'-  number: x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2-    + x1*x5 + x2*x5 + x3*x5 + x4*x5 + x5^2 + x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6-    + x6^2-- params:-    n: '6'-    k: '3'-  number: x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2-    + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4-    + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3 + x1^2*x5 + x1*x2*x5 + x2^2*x5 + x1*x3*x5-    + x2*x3*x5 + x3^2*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x4^2*x5 + x1*x5^2 + x2*x5^2-    + x3*x5^2 + x4*x5^2 + x5^3 + x1^2*x6 + x1*x2*x6 + x2^2*x6 + x1*x3*x6 + x2*x3*x6-    + x3^2*x6 + x1*x4*x6 + x2*x4*x6 + x3*x4*x6 + x4^2*x6 + x1*x5*x6 + x2*x5*x6 + x3*x5*x6-    + x4*x5*x6 + x5^2*x6 + x1*x6^2 + x2*x6^2 + x3*x6^2 + x4*x6^2 + x5*x6^2 + x6^3+  '1':+    '1': x1+    '2': x1^2+    '3': x1^3+  '2':+    '1': x1 + x2+    '2': x1^2 + x1*x2 + x2^2+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3+  '3':+    '1': x1 + x2 + x3+    '2': x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2+      + x2*x3^2 + x3^3+  '4':+    '1': x1 + x2 + x3 + x4+    '2': x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2+      + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4+      + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3+  '5':+    '1': x1 + x2 + x3 + x4 + x5+    '2': x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2+      + x1*x5 + x2*x5 + x3*x5 + x4*x5 + x5^2+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2+      + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4+      + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3 + x1^2*x5 + x1*x2*x5 + x2^2*x5 + x1*x3*x5+      + x2*x3*x5 + x3^2*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x4^2*x5 + x1*x5^2 ++      x2*x5^2 + x3*x5^2 + x4*x5^2 + x5^3+  '6':+    '1': x1 + x2 + x3 + x4 + x5 + x6+    '2': x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2+      + x1*x5 + x2*x5 + x3*x5 + x4*x5 + x5^2 + x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6+      + x6^2+    '3': x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2+      + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4+      + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3 + x1^2*x5 + x1*x2*x5 + x2^2*x5 + x1*x3*x5+      + x2*x3*x5 + x3^2*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x4^2*x5 + x1*x5^2 ++      x2*x5^2 + x3*x5^2 + x4*x5^2 + x5^3 + x1^2*x6 + x1*x2*x6 + x2^2*x6 + x1*x3*x6+      + x2*x3*x6 + x3^2*x6 + x1*x4*x6 + x2*x4*x6 + x3*x4*x6 + x4^2*x6 + x1*x5*x6 ++      x2*x5*x6 + x3*x5*x6 + x4*x5*x6 + x5^2*x6 + x1*x6^2 + x2*x6^2 + x3*x6^2 + x4*x6^2+      + x5*x6^2 + x6^3