[ui] fix searching for multi partitions (#26580)

## Summary & Motivation
Linear:
https://linear.app/dagster-labs/issue/FE-735/bug-partition-filter-in-ui-for-multipartitionsdefinition-f

Make searchValues state an array of searchValue

## How I Tested These Changes
Tested with multi_partitions asset with app-proxy to dogfood-test-1

js_modules/dagster-ui/packages/ui-core/src/assets/AssetPartitions.tsx

@@ -94,10 +94,20 @@ export const AssetPartitions = ({
         .filter((s: AssetPartitionStatus) => DISPLAYED_STATUSES.includes(s)),
   });
 
-  const [searchValue, setSearchValue] = useQueryPersistedState<string>({
-    queryKey: 'search',
-    defaults: {search: ''},
-  });
+  const [searchValues, setSearchValues] = useState<string[]>([]);
+  const updateSearchValue = (idx: number, value: string) => {
+    setSearchValues((prev) => {
+      const next = [...prev];
+
+      // add empty strings for missing indices
+      while (next.length <= idx) {
+        next.push('');
+      }
+
+      next[idx] = value;
+      return next;
+    });
+  };
 
   // Determine which axis we will show at the top of the page, if any.
   const timeDimensionIdx = selections.findIndex((s) => isTimeseriesDimension(s.dimension));
@@ -148,7 +158,7 @@ export const AssetPartitions = ({
     const sortType = getSort(sortTypes, idx, selections[idx]!.dimension.type);
 
     // Apply the search filter
-    const searchLower = searchValue.toLocaleLowerCase().trim();
+    const searchLower = searchValues?.[idx]?.toLocaleLowerCase().trim() || '';
     const filteredKeys = allKeys.filter((key) => key.toLowerCase().includes(searchLower));
 
     const getSelectionKeys = () =>
@@ -269,9 +279,10 @@ export const AssetPartitions = ({
                   <TextInput
                     fill
                     icon="search"
-                    value={searchValue}
-                    onChange={(e) => setSearchValue(e.target.value)}
+                    value={searchValues[idx] || ''}
+                    onChange={(e) => updateSearchValue(idx, e.target.value)}
                     placeholder="Filter by name…"
+                    data-testId={testId(`search-${idx}`)}
                   />
                 </Box>
                 <div>

js_modules/dagster-ui/packages/ui-core/src/assets/__tests__/AssetPartitions.test.tsx

@@ -242,4 +242,115 @@ describe('AssetPartitions', () => {
     });
     expect(screen.queryByTestId('dimension-range-input')).toBeNull();
   });
+
+  it('should support searching within a multiple asset partition', async () => {
+    const Component = () => {
+      const [params, setParams] = useState<AssetViewParams>({});
+      return (
+        <MemoryRouter>
+          <MockedProvider mocks={[MultiDimensionTimeFirstPartitionHealthQuery]}>
+            <AssetPartitions
+              assetKey={{path: ['multi_dimension_time_first']}}
+              params={params}
+              setParams={setParams}
+              paramsTimeWindowOnly={false}
+              assetPartitionDimensions={['date', 'zstate']}
+              dataRefreshHint={undefined}
+              isLoadingDefinition={false}
+            />
+          </MockedProvider>
+        </MemoryRouter>
+      );
+    };
+
+    render(<Component />);
+
+    await waitFor(() => {
+      expect(screen.getByTestId('partitions-date')).toBeVisible();
+      expect(screen.getByTestId('partitions-zstate')).toBeVisible();
+    });
+
+    await waitFor(() => {
+      expect(
+        getByTestId(
+          screen.getByTestId('partitions-date'),
+          'asset-partition-row-2023-02-05-index-0',
+        ),
+      ).toBeVisible();
+      expect(
+        getByTestId(screen.getByTestId('partitions-zstate'), 'asset-partition-row-TN-index-0'),
+      ).toBeVisible();
+    });
+
+    // search partitions
+    const searchInput = screen.getByTestId(`search-0`);
+    await userEvent.type(searchInput, '2023-01-3');
+
+    // verify partitions search results
+    await waitFor(() => {
+      expect(
+        getByTestId(
+          screen.getByTestId('partitions-date'),
+          'asset-partition-row-2023-01-31-index-0',
+        ),
+      ).toBeVisible();
+      expect(
+        getByTestId(
+          screen.getByTestId('partitions-date'),
+          'asset-partition-row-2023-01-30-index-1',
+        ),
+      ).toBeVisible();
+      expect(
+        screen.queryByTestId('asset-partition-row-2023-02-05-index-0'),
+      ).not.toBeInTheDocument();
+
+      // verify zstate is unchanged
+      expect(
+        getByTestId(screen.getByTestId('partitions-zstate'), 'asset-partition-row-TN-index-0'),
+      ).toBeVisible();
+    });
+
+    // search zstate
+    const searchInput1 = screen.getByTestId(`search-1`);
+    await userEvent.type(searchInput1, 'VA');
+
+    await waitFor(() => {
+      // verify zstate search filters properly
+      expect(
+        getByTestId(screen.getByTestId('partitions-zstate'), 'asset-partition-row-VA-index-0'),
+      ).toBeVisible();
+      expect(screen.queryByTestId('asset-partition-row-TN-index-0')).not.toBeInTheDocument();
+
+      // verify partitions search results are unchanged
+      expect(
+        getByTestId(
+          screen.getByTestId('partitions-date'),
+          'asset-partition-row-2023-01-31-index-0',
+        ),
+      ).toBeVisible();
+      expect(
+        getByTestId(
+          screen.getByTestId('partitions-date'),
+          'asset-partition-row-2023-01-30-index-1',
+        ),
+      ).toBeVisible();
+    });
+
+    // clear the search input
+    await userEvent.clear(searchInput);
+    await userEvent.clear(searchInput1);
+
+    // verify original rows are visible again
+    await waitFor(() => {
+      expect(
+        getByTestId(
+          screen.getByTestId('partitions-date'),
+          'asset-partition-row-2023-02-05-index-0',
+        ),
+      ).toBeVisible();
+      expect(
+        getByTestId(screen.getByTestId('partitions-zstate'), 'asset-partition-row-TN-index-0'),
+      ).toBeVisible();
+    });
+  });
 });